Go Math Answer Key

Go Math Grade 4 Chapter 5 Answer Key Pdf: Contains Factors, Common factors, Divisibilities and Review tests, etc. Which helps students for solving assignments and also for preparing in exams. In this chapter each and every question was explained in a simple way by which students can understand easily. Go Math Grade 4 Answer Key explains each question in a unique and simple way by that students can easily understand the solution.

Factors, Multiples, and Patterns Go Math Grade 4 Chapter 5 Answer Key Pdf

This Go Math Grade 4 Answer Key Chapter 5 Factors, Multiples, and Patterns have Grade 4 solutions that can be understood by anyone. Students, teachers can feel comfortable by this Go Math Grade 4 Answer Key as every question was solved in a simple way. By that students will not face any difficulty in understanding the solution.

Lesson 1: Model Factors

• Common Core – Model Factors – Page No. 283
• Common Core – Factors – Page No. 284
• Page No. 287
• Page No. 288

Lesson 2: Factors and Divisibility

• Common Core – Factors and Divisibility – Page No. 289
• Common Core – Factors – Page No. 290
• Page No. 293
• Page No. 294

Lesson 3: Problem Solving • Common Factors

• Common Core – Common Factors – Page No. 295
• Common Core – Common Factors – Page No. 296
• Page No. 297
• Page No. 298
• Page No. 301
• Page No. 302

Lesson 4: Factors and Multiples

• Common Core – Factors and Multiples – Page No. 303
• Common Core – Factors and Multiples – Page No. 304
• Page No. 307
• Page No. 308

Lesson 5: Prime and Composite Numbers

• Common Core – Prime and Composite Numbers – Page No. 309
• Common Core – Prime and Composite Numbers – Page No. 310
• Page No. 313
• Page No. 314

Lesson 6: Algebra • Number Patterns

• Common Core – Number Patterns – Page No. 315
• Common Core – Number Patterns – Page No. 316

Chapter 5 Review/Test

• Review/Test – Page No. 317
• Review/Test – Page No. 318
• Review/Test – Page No. 319
• Review/Test – Page No. 320
• Review/Test – Page No. 321
• Review/Test – Page No. 322
• Page No. 329
• Page No. 330

Common Core – Model Factors – Page No. 283

Model Factors

Use tiles to find all the factors of the product.

Record the arrays on grid paper and write the factors shown.

Question 1.

Question 2.
Write the factors of 30

Answer:
The Factors Of 30 are 1,2,3,5,6,10,15,30.

Explanation:
Factors are the numbers which divides the original number completely. Here, we can see the numbers which gives the result as 30 when multiplied together.So the factors of 30 are 1,2,3.5,6,10,15,30.

1×30=30
2×15=30
3×10=30
5×6=30
6×5=30
10×3=30
15×2=30
30×1=30

Question 3.
Write the factors of: 45

Answer: The Factors Of 45 are:1,3,5,9,15,45.

Explanation:
Factors are the numbers which divides the original number completely. Here, we can see the numbers which gives the result as 45 when multiplied together.So the factors of 45 are:1,3,5,9,15,45.

1×45=45
3×15=45
5×9=45
9×5=45
15×3=45
45×1=45

Go Math Grade 4 Lesson 5.1 Answer Key Question 4.
Write the factors of 19

Answer: The Factors Of 19 are:1,19.

Explanation:
Since 19 is a Prime number that means it is divisible by 1 and itself. So the factors of  19 are 1,19.

1×19=19
19×1=19.

Question 5.
Write the factors of: 40

Answer: The Factors Of 40 are:1,2,4,5,8,10,20,40.

Explanation:Factors are the numbers which divides the original number completely. The Factors Of 40 are:1,2,4,5,8,10,20,40.

1×40=40
2×20=40
4×10=40
5×8=40
8×5=40
10×4=40
20×2=40
40×1=40

Question 6.
Write the factors of: 36

Answer: The Factors Of 36 are:1,2,3,4,6,9,12,18,36.

Explanation:
Factors are the numbers which divides the original number completely. The factors of 36 are:1,2,3,4,6,9,12,18,36.

1×36=36
2×18=36
3×12=36
4×9=36
6×6=36
9×4=36
12×3=36
18×3=36
36×1=36.

Question 7.
Write the factors of: 22

Answer: The Factors Of 22 are:1,2,11,22.

Explanation:
Factors are the numbers which divides the original number completely. The factors of 22 are:1,2,11,22.

1×22=22
2×11=22
11×2=22
22×1=22.

Question 8.
Write the factors of: 4

Answer: The Factors Of 4 are:1,2,4.

Explanation:
Factors are the numbers which divides the original number completely. The Factors Of 4 are:1,2,4.

1×4=4
2×2=4
4×1=4.

Question 9.
Write the factors of: 26

Answer: The Factors Of 26 are:1,2,13,26.

Explanation:
Factors are the numbers which divides the original number completely. Here, we can see the numbers which gives the result as 26 when multiplied together.So the factors of 26 are:1,2,13,26.

1×26=26
2×13=26
13×2=26
26×1=26.

Go Math Grade 4 Chapter 5 Test Question 10.
Write the factors of: 49

Answer: The Factors Of 49 are:1,7,49.

Explanation:
Factors are the numbers which divides the original number completely. The Factors Of 49 are:1,7,49.

1×49=49
7×7=49
49×1=49.

Question 11.
Write the factors of: 32

Answer: The Factors Of 32 are:1,2,4,8,16,32.

Explanation:
Factors are the numbers that divide the original number completely. Here, we can see the numbers which give the result as 32 when multiplied together.So the factors of 32 are:1,2,4,8,16,32.

1×32=32
2×16=32
4×8=32
8×4=32
16×2=32
32×1=32.

Question 12.
Write the factors of 23

Answer: The Factors Of 23 are:1,23.

Explanation:
Since 23 is a Prime number that means it is divisible by 1 and itself. So the factors of  23 are 1,23.

1×23=23
23×1=23.

Lesson 1 Factors and Multiples Answer Key Question 13.
Brooke has to set up 70 chairs in equal rows for the class talent show. But, there is no room for more than 20 rows. What are the possible number of rows that Brooke could set up?

Answer:
The answer is 2,5,7,10,14.

Explanation:
Let the possible no.of rows be X, As there is no room for more than 20 rows so there should not be more than 20 rows.X should be less than or equal to 20(X<=20). As Brooke has 70 chairs to set up in equal rows we will find the factors of 70 and in that, we must pick up the numbers which are less than equal to 20. Therefore the factors of 70 are 2,5,7,10,14.

Question 14.
Eduardo thinks of a number between 1 and 20 that has exactly 5 factors. What number is he thinking of?

Answer: 16

Explanation: If find factors for 1 to 20 we don’t get exactly 5 factors for any number except 16. So the answer is 16.

Common Core – Factors – Page No. 284

Lesson Check

Question 1.
Which of the following lists all the factors of 24?
Options:
a. 1, 4, 6, 24
b. 1, 3, 8, 24
c. 3, 4, 6, 8
d. 1, 2, 3, 4, 6, 8, 12, 24

Answer: d(1, 2, 3, 4, 6, 8, 12, 24)

Explanation:Factors are the numbers which divides the original number completely. Here, we can see the numbers which gives the result as 24 when multiplied together. So the factors of 24 are:1, 2, 3, 4, 6, 8, 12, 24.

1×24=24
2×12=24
3×8=24
4×6=24
6×4=4
8×3=24
12×2=24
24×1=24

Question 2.
Natalia has 48 tiles. Which of the following shows a factor pair for the number 48?
Options:
a. 4 and 8
b. 6 and 8
c. 2 and 12
d. 3 and 24

Answer: b(6 and 8)

Explanation: 6 and 8 are factor pair for 48 because 6×8=48.

Spiral Review

Question 3.
The Pumpkin Patch is open every day. If it sells 2,750 pounds of pumpkins each day, about how many pounds does it sell in 7 days?
Options:
a. 210 pounds
b. 2,100 pounds
c. 14,000 pounds
d. 21,000 pounds

Answer: d

Explanation: Let’s round off 2750 pounds to 3000 pounds. In one day 3000 pounds of pumpkins were sold out, and in
7 days?? —- 3000×7= 21,000 pounds.

Question 4.
What is the remainder in the division problem modeled below?

Options:
a. 2
b. 3
c. 5
d. 17

Answer: a

Explanation: We can see in the above figure 3 circles with 5 sub circles inside it and a pair of  sub circles. Here total sub circles are (3×5)+2=17. If we divide 17 with 3 then we will get reminder as 2. So answer is 2.

Question 5.
Which number sentence is represented by the following array?

Options:
a. 4 × 5 = 20
b. 4 × 4 = 16
c. 5 × 2 = 10
d. 5 × 5 = 25

Answer: a

Explanation: As we can see 4 rows and 5 squares, So 4 × 5 = 20.

Go Math Grade 4 Chapter 5 Test Pdf Question 6.
Channing jogs 10 miles a week. How many miles will she jog in 52 weeks?
Options:
a. 30 miles
b. 120 miles
c. 200 miles
d. 520 miles

Answer: d

Explanation: No.of weeks = 52. So 1 week = 10 miles, then 52 weeks =?????
52×10=520 miles.

Page No. 287

Question 1.
Is 4 a factor of 28? Draw a model to help.
Think: Can you make a rectangle with 28 squares in 4 equal rows?

4 ______ a factor of 28.
Type below:
__________

Is 5 a factor of the number? Write yes or no.

Question 2.
27
Answer: No.

Explanation: Factors of 27 are 1,3,9,27. So the answer is No.

Question 3.
30
Answer : Yes.

Explanation: As the last digit is 0 which is divisible 5.

Question 4.
36
Answer: No

Explanation: 36 is not divisible by 5, So the answer is no

Question 5.
53
Answer: No

Explanation: Factors of 53 are 1, 53. So the answer is No.

Is 9 a factor of the number? Write yes or no.

Question 6.
54
Answer: Yes.

Explanation: As 54 is divisible by 9.

Question 7.
63
Answer: Yes.

Explanation: 63 is divisible by 9, So the answer is Yes

Question 8.
67
Answer: No.

Explanation: 67 is a prime number which means it is divisible by 1 and itself. So the answer is No.

Question 9.
93
Answer: No.

Explanation: The factors of 93 are 1,3,31 and 93. So the answer is No.

List all the factor pairs in the table.

Question 10.

Answer:
1×24=24    1,24
2×12=24     2,12
3×8=24       3,8
4×6=24       4,6

Explanation: Factors of 24.

Question 11.

Answer:
1×39=39    1,39
3×13=39.   3,13

Explanation: Factors of 39.

Practice: Copy and Solve List all the factor pairs for the number. Make a table to help.

Question 12.
56
Answer:
1×56=56     1,56
2×23=56     2,23
4×14=56      4,14
7×8=56         7,8
8×7=56         8,7

Explanation: Factors of 56.

Question 13.
64
Answer:
1×64=64    1,64
2×32=64    2,32
4×16=64    4,16
8×8=64      8,8

Explanation: Factors of 64 and factor pair for 64 is 8,8.

Page No. 288

Use the table to solve 14–15.

Question 14.
Dirk bought a set of stamps. The number of stamps in the set he bought is divisible by 2, 3, 5, 6, and 9. Which set is it?
Answer: 90

Explanation: 90 is divisible by all numbers 2,3,5,6, and 9. So the answer is 90.

Question 15.
Geri wants to put 6 stamps on some pages in her stamp book and 9 stamps on other pages. Explain how she could do this with the stamp set for Sweden.

Answer: 10 pages with 6 stamps and 2 pages with 9 stamps.

Explanation: Geri could break 78 into 60+18, As 60 is divisible by 6, and 18 is divisible by 9. Then she could make 10 pages with 6 stamps as 60÷6=10 and 2 pages with 9 stamps as 18÷9=2.

Question 16.
Use Counterexamples George said if 2 and 4 are factors of a number, then 8 is a factor of the number. Is he correct? Explain.

Answer: No

Explanation: Because if we 12 as an example, 2 and 4 are factors of 12 but not 8.

Question 17.
Classify the numbers. Some numbers may belong in more than one box.

Answer:
Divisible by 5 and 9 — 45
Divisible by 3 and 9 — 27,45,54,72,81
Divisible by 2 and 6 — 54,72,84.

Common Core – Factors and Divisibility – Page No. 289

Is 6 a factor of the number? Write yes or no.

Question 1.

Question 2.
56
Answer: No

Explanation: 56 is not divisible by 6. So the answer is No.

Question 3.
42

Answer: Yes

Explanation: Since 42 is divisible by 6.

Question 4.
66
Answer: Yes

Explanation: 66 is divisible by 6.

Is 5 a factor of the number? Write yes or no.

Question 5.
38
Answer: No

Explanation: If the end is 0 or 5 then the number is divisible by 5. As the number is 38 the answer is No

Question 6.
45

Answer: Yes

Explanation: 45 is divisible by 5. So the answer is Yes

Practice and Homework Lesson 5.2 Answer Key 4th Grade Question 7.
60
Answer: Yes

Explanation: 60 is a factor of 5 because 60 is divisible by 5.

Question 8.
39
Answer: No

Explanation: As 39 is not divisible by 5. So the answer is No.

List all the factor pairs

Question 9.
Factors of 12

Answer:
1 × 12 = 12; ( 1 , 12 )
2 × 6 = 12; ( 2, 6 )
3 × 4 = 12; ( 3 , 4 )

Question 10.
Factors of 25

Answer:
1 ×25  = 25; ( 1 , 25 )
5 × 5 = 25; ( 5 , 5 )

Question 11.
List all the factor pairs for 48.

Answer: Factor pairs of 48 are (1,48),(2,24),(3,16),(4,12),(6,8),(12,2),(6,3),(24,2),(48,1).

Explanation: Factor pairs are the pairs when we multiply both numbers will get the result. Here factor pairs for 48 are
1×48=48     (1,48)
2×24=48      (2,24)
3×16=48      (3,16)
4×12=48      (4,12)
6×8 =48       (6,8)

Problem Solving

Question 12.

Bryson buys a bag of 64 plastic miniature dinosaurs. Could he distribute them equally into six storage containers and not have any left over?

Answer: No

Explanation: 64 is not divisible by 6, So he cannot distribute them equally into six storage containers.

Question 13.
Lori wants to distribute 35 peaches equally into baskets. She will use more than 1 but fewer than 10 baskets. How many baskets does Lori need?

Answer: 5 or 7.

Explanation: First we need to know the factors of 35. The factors of 35 are 1,5,7,35. As Lori uses more than 1 but fewer than 10, the answer is 5 or 7. Lori can distribute 35 peaches equally in 5 or 7 baskets.

Common Core – Factors – Page No. 290

Lesson Check

Question 1.
Which of the following numbers has 9 as a factor?
Options:
a. 28
b. 30
c. 39
d. 45

Answer: d

Explanation: 45 is divisible 9. So the answer is 45.

Question 2.
Which of the following numbers does NOT have 5 as a factor?
Options:
a. 15
b. 28
c. 30
d. 45

Answer: 28

Explanation: 28 is not divisible by 5. So 28 is not a factor of 5.

Spiral Review

Question 3.
Which of the following shows a strategy to use to find 4 × 275?
Options:
a. (4 × 300) + (4 × 25)
b. (4 × 300) – (4 × 25)
c. (4 × 275) – 100
d. (4 × 200) + 75

Answer: b

Explanation: First we must replace 300-25 in the place of 275 then it becomes 4×(300-25), Now we must use the distributive property of multiplication then (4×300)-(4×25). So the answer is b.

Question 4.
Jack broke apart 5 × 216 as (5 × 200) + (5 × 16) to multiply mentally. What strategy did Jack use?
Options:
a. the Commutative Property
b. the Associative Property
c. halving and doubling
d. the Distributive Property

Answer: d

Explanation: Distributive property means if we multiply a sum by a number is same as multiplying each addend by the number and adding the products. This is the strategy Jack used.

Question 5.
Jordan has $55. She earns $67 by doing chores. How much money does Jordan have now?
Options:
a. $122
b. $130
c. $112
d. $12

Answer: a

Explanation: Jordan has $55, she earns by doing chores is $67. So total money is $55+$67=$122.

Question 6.
Trina has 72 collector’s stamps. She puts 43 of the stamps into a stamp book. How many stamps are left?
Options:
a. 29
b. 31
c. 39
d. 115

Answer: a

Explanation: Stamps left are 72-43=29.

Page No. 293

Question 1.
Lucy has 40 bean plants, 32 tomato plants, and 16 pepper plants. She wants to put the plants in rows with only one type of plant in each row. All rows will have the same number of plants. How many plants can Lucy put in each row?
First, read the problem and think about what you need to find. What information will you use? How will you use the information?

Answer: We will find common factors for 40,32 and 16.

Question 1.
Next, make a list. Find the factors for each number in the problem.

Answer:
Factors of 40 are — 1,2,4,5,8,10,20,40
Factors of 32 are — 1,2,4,8,16,32
Factors of 16 are — 1,2,4,8,16

Question 1.
Finally, use the list. Circle the common factors.
So, Lucy can put ___ , ___ , ___ , or ___ plants in each row.

Answer: 1,2,4,8

Explanation: Because 1,2,4,8, are common factors in 40,32,16.

Question 2.
What if Lucy has 64 bean plants instead of 40 bean plants? How many plants can Lucy put in each row?

Answer: 1,2,4,8,16

Explanation: Here we need to find the factors of 64,32 and 16. We get common factors as 1,2,4,8,16.

Go Math Grade 4 Lesson 5.2 Answer Key Question 3.
One common factor of two numbers is 40. Another common factor is 10. If both numbers are less than 100, what are the two numbers?
______ and ______

Answer:  40 and 80.

Explanation: As the next multiple of 40 is 80. So both 40 and 80 are less than 100 and has a common factor as 10.

Question 4.
The sum of two numbers is 136. One number is 51. What is the other number? What are the common factors of these two numbers?

Answer: 85.
Common Factors are 1,17.

Explanation: As 136-51= 85
Factors of 51 are 1,3,17,51
Factors of 85 are 1,5,17,85.

Page No. 294

Question 5.
Analyze A number is called a perfect number if it equals the sum of all of its factors except itself. For instance, 6 is a perfect number because its factors are 1, 2, 3, and 6, and 1 + 2 + 3 = 6. What is the next greater perfect number?

Answer: 28

Explanation: The factors of 28 are 1,2,4,7,14 and 28. If we add 1+2+4+7+14 we will get 28. So 28 is a perfect number.

Question 6.
Sona knits 10 squares a day for 7 days. Can she sew together the squares to make 5 equal-sized blankets? Explain.

Answer: Yes

Explanation: As 10×7= 70 which is a factor of 5.

Question 7.
Julianne earned $296 working at a grocery store last week. She earns $8 per hour. How many hours did Julianne work?
Answer: 37 hours

Explanation: Julianne earned $296 in last week. Per hour she earns $8, So total no.of hours did she worked is
296÷8= 37 hours.

Question 8.
There are 266 students watching a play in the auditorium. There are 10 rows with 20 students in each row and 5 rows with 8 students in each row. How many students are sitting in each of the 2 remaining rows if each of those rows has an equal number of students?

Answer: 13 Students

Explanation: Total number of students is 266. In which 10 rows were filled with 20 students that means 10×20=200 students, and 5 rows were filled with 8 students which means 5×8= 40 students. The total students filled are 240. And to know how many students filled in the remaining 2 rows we need to subtract 266-240=26, As students are filled in 2 rows 26÷2= 13.

Question 9.
Ben is planting a garden with 36 zinnias, 18 marigolds, and 24 petunias. Each row will have only one type of plant. Ben says he can put 9 plants in each row. He listed the common factors of 36, 18 and 24 below to support his reasoning.
36: 1, 2, 3, 4, 6, 9, 12, 18, 36
18: 1, 2, 3, 6, 8, 9, 18
24: 1, 2, 3, 4, 6, 8, 9, 12, 24
Is he correct? Explain your answer. If his reasoning is incorrect, explain how he should have found the answer.

Answer: No

Explanation: The factors of 18 and 24 are incorrect which he listed. And the common factors for 36,24 and 18 are 1,2,3 and 6. So he can put 1,2,3 and 6 plants in a row.

Common Core – Common Factors – Page No. 295

Problem Solving Common Factors

Solve each problem.

Question 1.
Grace is preparing grab bags for her store’s open house. She has 24 candles, 16 pens, and 40 figurines. Each grab bag will have the same number of items, and all the items in a bag will be the same. How many items can Grace put in each bag?

Question 2.
Simon is making wreaths to sell. He has 60 bows, 36 silk roses, and 48 silk carnations. He wants to put the same number of items on each wreath. All the items on a wreath will be the same type. How many items can Simon put on each wreath?

Answer:1,2,3,4,6 or 12 items Simon puts on each wreath.

Explanation: First we will find the common factors of 36,48,60
factors of 36 are: 1,2,3,4,6,9,12,18,36.
factors of 48 are: 1,2,3,4,6,8,12,16,24,48
factors of 60 are: 1,2,3,4,5,6,10,12,15,20,30,60.
The common factors of 36,48,60 are 1,2,3,4,6,12. So Simon can put 1,2,3,4,6 or 12 items on each wreath.

Question 3.
Justin has 20 pencils, 25 erasers, and 40 paper clips. He organizes them into groups with the same number of items in each group. All the items in a group will be the same type. How many items can he put in each group?

Answer: Justin can put 1 or 5 items in each group.

Explanation: We will find common factors of 20,25,40.
factors of 20 are: 1,2,4,5,10,20.
factors of 25 are: 1,5,25.
factors of 40 are: 1,2,4,5,8,10,20,40
So common factors are 1 and 5.

Question 4.
A food bank has 50 cans of vegetables, 30 loaves of bread, and 100 bottles of water. The volunteers will put the items into boxes. Each box will have the same number of food items and all the items in the box will be the same type. How many items can they put in each box?

Answer: 1,2,5, or 10.

Explanation: 1,2,5 or 10 are the common factors of 30,50 and 100.
factors for 30 are: 1,2,3,5,6,10,15,30
factors for 50 are: 1,2,5,10,25,50
factors of 100 are: 1,2,4,5,10,20,25,50,100
So answer is 1,2,5,10.

Go Math Grade 4 Chapter 5 Test Answer Key Question 5.
A debate competition has participants from three different schools: 15 from James Elementary, 18 from George Washington School, and 12 from the MLK Jr. Academy. All teams must have the same number of students. Each team can have only students from the same school. How many students can be on each team?

Answer: 3

Explanation: Lets find the common factors of 12,15,18
factors of 12 are: 1,2,3,4,6,12
factors of 15 are: 1,3,5,15
factors of 18 are: 1,2,3,6,9,18
3 is the common factor for 12,15,18

Common Core – Common Factors – Page No. 296

Lesson Check

Question 1.
What are all the common factors of 24, 64, and 88?
Options:
a. 1 and 4
b. 1, 4, and 8
c. 1, 4, 8, and 12
d. 1, 4, 8, and 44

Answer: b

Explanation:
factors of 24 are: 1,2,3,4,8,12,24
factors of 64 are: 1,2,4,8,16,32,64
factors of 88 are: 1,2,4,8,11,22,44,88

Question 2.
Which number is NOT a common factor of 15, 45, and 90?
Options:
a. 3
b. 5
c. 10
d. 15

Answer: c

Explanation: As 15 and 45 are not divisible by 10.

Spiral Review

Question 3.
Dan puts $11 of his allowance in his savings account every week. How much money will he have after 15 weeks?
Options:
a. $165
b. $132
c. $110
d. $26

Answer: a

Explanation: Dan puts $11 in his savings account every week, So after 15 weeks it will be 15×11=165.
The total money he will have after 15 weeks is $165.

Question 4.
James is reading a book that is 1,400 pages. He will read the same number of pages each day. If he reads the book in 7 days, how many pages will he read each day?
Options:
a. 20
b. 50
c. 140
d. 200

Answer: d

Explanation: Total no.of.pages is 1400, no.of pages James read each day is 1400÷7= 200

Question 5.
Emma volunteered at an animal shelter for a total of 119 hours over 6 weeks. Which is the best estimate of the number of hours she volunteered each week?
Options:
a. 10 hours
b. 20 hours
c. 120 hours
d. 714 hours

Answer: b

Explanation: Total hours Emma volunteered is 119 hours over 6 weeks, how much she volunteered each week is
119÷6= 19.833 i.e 20 hours. We must round off to the nearest one i.e 20 hours.

Question 6.
Which strategy can be used to multiply 6 × 198 mentally?
Options:
a. 6 × 198 = (6 × 19) + (6 × 8)
b. 6 × 198 = (6 × 200) + (6 × 2)
c. 6 × 198 = (6 × 200) – (6 × 2)
d. 6 × 198 = (6 + 200) × (6 + 2)

Answer: c

Explanation: By Distributive property of multiplication 6×198 can be written as (6 × 200) – (6 × 2).

Page No. 297

Choose the best term from the box.

Question 1.
A number that is multiplied by another number to find a product is called a

Answer: Factor.

Question 2.
A number is _________ by another number if the quotient is a counting number and the remainder is zero.
Answer: Divisible.

List all the factors from least to greatest.

Question 3.
8
Answer: 1,2,4,8

Question 4.
14
Answer: 1,2,7,14

Is 6 a factor of the number? Write yes or no.

Question 5.
81
Answer: No

Explanation: 81 is not divisible by 6

Question 6.
45
Answer: No

Explanation: 45 is not divisible by 6

Question 7.
42
Answer: Yes

Explanation: 42 is divisible by 6

Question 8.
56

Answer: No.

Explanation: 56 is not divisible by 6

List all the factor pairs in the table.

Question 9.

Answer:
1×64=64    1,64
2×32=64    2,32
4×16=64    4,16
8×8=64      8,8

Explanation: Factors of 64

Question 10.

Answer:
1×44=44    1,44
2×22=44    2,22
11×4=44    11,4

List the common factors of the numbers.

Question 11.
9 and 18

Answer: 1,3,9

Explanation:
Factors of 9 are: 1,3,9
Factors of 18 are: 1,2,3,9,18

Question 12.
20 and 50

Answer: 1,2,5,10

Explanation:
Factors of 20 are: 1,2,4,5,10,20
Factors of 50 are: 1,2,5,10,25,50

Page No. 298

Question 13.
Sean places 28 tomato plants in rows. All rows contain the same number of plants. There are between 5 and 12 plants in each row. How many plants are in each row?

Answer: 7 plants.

Explanation: There are 28 tomato plants in a row. To find out how many plants in a row we will find the factors of 28 i.e 1,2,4,7,14,28. As there are between 5 and 12 plants 7 is the only number between 5 and 12. So 7 plants are planted in each row.

Question 14.
Ella bought some key chains and spent a total of $24. Each key chain costs the same whole-dollar amount. She bought between 7 and 11 key chains. How many key chains did Ella buy?

Answer: 8

Explanation: Ella spent a total of $24. To find how many key chains first we will find the factors of 24. Factors of 24 are
1,2,3,4,6,8,12,24. As Ella bought between 7 and 11 key chains 8 is the only number between 7 and 11. So 8 key chains Ella bought.

Question 15.
Sandy has 16 roses, 8 daisies, and 32 tulips. She wants to arrange all the flowers in the bouquets. Each bouquet has the same number of flowers and the same type of flower. What is the greatest number of flowers that could be in a bouquet?

Answer: 2 roses, 1 daisy, and 4 tulips in 8 bouquets.

Explanation: First we must add all the flowers i.e 16+8+32= 56, Now we can divide 56 flowers equally in each bouquet. Like 2 roses, 1 daisy and 4 tulips in 8 bouquets or 8 roses in 2 bouquets, 8 daisies in 1 bouquet and 8 tulips in 4 bouquets.

Question 16.
Amir arranged 9 photos on a bulletin board. He put the photos in rows. Each row contains the same number of photos. How many photos could be in each row?

Answer: 9 photos in a row and 3 photos in 3 rows or 9 photos in 1 row.

Explanation: Factors of 9 are 1,3,9. So Amir can arrange 9 photos in a row and 3 photos in 3 rows or 9 photos in 1 row.

Page No. 301

Question 1.
Multiply to list the next five multiples of 4.
4 , _____ , _____ , _____ , _____ , _____
1 × 4
4 , _____ , _____ , _____ , _____ , _____

Answer:
4     1×4
8     2×4
12   3×4
16   4×4
20   4×5

Explanation: Multiplies of 4

Is the number a factor of 6? Write yes or no.

Question 2.
2

Answer: Yes

Explanation: 6 is divisible by 2. So 2 is the factor of 6.

Question 3.
6

Answer: Yes

Explanation: 6 is divisible by 6.

Question 4.
16

Answer: No

Explanation: 16 is not divisible by 6

Question 5.
18

Answer: Yes

Explanation: 18 is divisible by 6

Is the number a multiple of 6? Write yes or no.

Question 6.
3

Answer: No

Explanation: Multiples of 6 are 6,12,18,24,30, etc.

Question 7.
6

Answer: Yes

Explanation: 1×6= 6. So 6 is multiple of 6.

Question 8.
16

Answer: No

Explanation: Multiples of 6 are 6,12,18,24,30, etc.

Question 9.
18

Answer: Yes

Explanation: Multiples of 6 are 6,12,18,24,30, etc.

Is the number a multiple of 3? Write yes or no.

Question 10.
4

Answer: No

Explanation: Multiples of 3 are 3,6,9,12,15,etc.

Question 11.
8

Answer: No

Explanation: Multiples of 3 are 3,6,9,12,15,etc.

Question 12.
24

Answer: Yes

Explanation: Multiples of 3 are 3,6,9,12,15,etc.

Question 13.
38

Answer: No

Explanation: Multiples of 3 are 3,6,9,12,15,18,21,24,27,30,33,36,39,42,etc.

Question 14.
List the next nine multiples of each number. Find the common multiples.
Multiples of 2: 2, _________________
Multiples of 8: 8, _________________
Common multiples: _________________

Answer: 8,16.

Explanation:
Multiples of 2: 2,4,6,8,10,12,14,16,18,20.
Multiples of 8: 8,16,24,32,40,48,56,64,72,80.
So common multiples are: 8,16

Generalize Algebra Find the unknown number.

Question 15.
12, 24, 36, _____

Answer: 48

Explanation:
12×1= 12
12×2= 24
12×3= 36
12×4= 28

Question 16.
25, 50, 75, 100, ______

Answer: 125

Explanation:
25×1= 25
25×2= 50
25×3= 75
25×4= 100
25×5= 125

Tell whether 20 is a factor or multiple of the number.

Write factor, multiple, or neither.

Question 17.
10

Answer: Multiple

Explanation: 2×10= 20.

Question 18.
20

Answer: Factor and multiple

Explanation:
1×20= 20
20÷1= 20.

Question 19.
30

Answer: Neither

Explanation:
Factors of 30 are:    1,2,3,5,6,10,15,and 30.
Multiples of 30 are: 30,60,90,etc.

Write true or false. Explain.

Question 20.
Every whole number is a multiple of 1.

Answer: True.

Explanation: For every whole number which is multiplied with 1, the result will be that number.

Question 21.
Every whole number is a factor of 1.

Answer: False

Explanation: Not every whole number is a factor of 1.

Go Math Chapter 5 Review Test 4th Grade Question 22.
Julio wears a blue shirt every 3 days. Larry wears a blue shirt every 4 days. On April 12, both Julio and Larry wore a blue shirt. What is the next date that they will both wear a blue shirt?

Answer: April 24

Explanation:
As Julio wears a blue shirt every 3 days and another shirt in the remaining 4 days, So 4×3 days= 12
Larry wears a blue shirt every 4 days and another shirt in the remaining 3 days, So 3×4 days= 12
12+12= 24. So the next date will be April 24.

Page No. 302

Complete the Venn diagram. Then use it to solve 23–25.

Question 23.
What multiples of 4 are not factors of 48?

Answer: 20,28,32,36,40,44.

Explanation:
Multiples of 4 are 4,8,12,16,20,24,28,32,36,40,44,48.
Not a factors of 48 are 20,28,32,36,40,44.

Question 24.
What factors of 48 are multiples of 4?

Answer: 4,8,12,16,24,48.

Explanation:
Multiples of 4 are: 4,8,12,16,20,24,28,32,36,40,44,48.
Factors of 48 are: 1,2,4,8,12,16,24,48.

Question 25.
Pose a Problem Look back at Problem 24. Write a similar problem by changing the numbers. Then solve.

Answer: Let’s take factors of 64 are multiples of 8?
8,16,32,64.

Explanation:
Multiples of 8 are: 8,16,24,32,40,48,56,64,72,80
Factors of 64 are: 1,2,4,8,16,32,64

Question 26.
Kia paid $10 for two charms. The price of each charm was a multiple of $2. What are the possible prices of the charms?

Answer: $2,$8 and $4,$6.

Explanation: Since the price was multiple of 2 and Kia paid $10 for two charms, So possible prices are $2+$8=$10
and $4+$6=$10.

Question 27.
Look for Structure The answer is 9, 18, 27, 36, 45. What is the question?

Answer: Write the multiples of 9

Question 28.
How do you know whether a number is a multiple of another number?

Answer: When the number is divisible by the number then that number is multiple of another number.

Explanation: For example, if we take a number i.e 8 which is divisible by 2 and 8 is a multiple of 2.

Question 29.
For numbers 29a–29e, select True or False for each statement.
a. The number 45 is a multiple of 9.
i. True
ii. False

Answer: True

Explanation: As 9×5= 45, So 45 is multiple of 9.

Question 29.
b. The number 4 is a multiple of 16.
i. True
ii. False

Answer: False.

Explanation: As 16 is divisible by 4 and not a multiple of 16.
Multiple of 16 are : 16,32,48,64,80.

Question 29.
c. The number 28 is a multiple of 4.
i. True
ii. False

Answer: True.

Explanation: 4×7=28.

Question 29.
d. The number 4 is a factor of 28.
i. True
ii. False

Answer: True.

Explanation:
Factors of 28 are: 1,2,4,7,14,28.

Question 29.
e. The number 32 is a factor of 8.
i. True
ii. False

Answer:

Explanation:

Common Core – Factors and Multiples – Page No. 303

Factors and Multiples
Is the number a multiple of 8? Write yes or no.

Question 1.

Question 2.
8
Answer: Yes

Explanation: Since 8×1=8, it is a multiple of 8

Question 3.
20
Answer: No

Explanation: 20 is not a multiple of 8

Question 4.
40
Answer: Yes

Explanation: 8×5=40, So 40 is multiple of 8

List the next nine multiples of each number. Find the common multiples.

Question 5.
Multiples of 4:
Multiples of 7:
Common multiples:

Answer:

Explanation:
Multiples of 4: 4,8,12,16,20,24,28,32,36,40.
Multiples of 7: 7,14,21,28,35,42,49,56,63,70.
Common Multiples: 28,

Question 6.
Multiples of 3:
Multiples of 9:
Common multiples:

Answer: 9,18,45,54,63, etc.

Explanation:
Multiples of 3: 3,6,9,12,15,18,21,24,27,30,33,36,39,42,45,48,51,54,57,60,63.
Multiples of 9: 9,18,27,36,45,54,63,72,81,90.
Common multiples: 9,18,45,54,63, etc.

Question 7.
Multiples of 6:
Multiples of 8:
Common multiples:

Answer: 24,48,72.

Explanation:
Multiples of 6: 6,12,18,24,30,36,42,48,54,60,66,72,78.
Multiples of 8: 8,16,24,32,40,48,56,64,72,80.
Common multiples: 24,48,72.

Tell whether 24 is a factor or multiple of the number. Write factor, multiple, or neither.

Question 8.
6

Answer: Multiple

Explanation: 6×4=24

Question 9.
36

Answer: Neither

Explanation: 36 is not a factor or multiple of 24.

Question 10.
48

Answer: Factor

Explanation: 24×2= 48, So 48 is a factor of 24

Problem Solving

Question 11.
Ken paid $12 for two magazines. The cost of each magazine was a multiple of $3. What are the possible prices of the magazines?

Answer: $3+$9=$12.

Explanation: As each magazine cost was multiple of $3, The possible price for 2 magazines are $3+$9=$12, which is a multiple of 3

Question 12.
Jodie bought some shirts for $6 each. Marge bought some shirts for $8 each. The girls spent the same amount of money on shirts. What is the least amount they could have spent?

Answer: $24

Explanation: As they spent the same amount of money which means the number should be multiple of $6 and $8, So multiples of 6 are: 6,12,18,24,30,36,42 and
multiples of 8 are: 8,16,24,32,40. The least amount they could spend is 24. As 24 is the least common multiple.

Common Core – Factors and Multiples – Page No. 304

Lesson Check

Question 1.
Which list shows numbers that are all multiples of 4?
Options:
a. 2, 4, 6, 8
b. 3, 7, 11, 15, 19
c. 4, 14, 24, 34
d. 4, 8, 12, 16

Answer: d

Explanation: Multiples of 4 are 4,8,12,16.

Question 2.
Which of the following numbers is a common multiple of 5 and 9?
Options:
a. 9
b. 14
c. 36
d. 45

Answer: 45

Explanation: 5×9= 45

Spiral Review

Question 3.
Jenny has 50 square tiles. She arranges the tiles into a rectangular array of 4 rows. How many tiles will be left over?
Options:
a. 0
b. 1
c. 2
d. 4

Answer: 2

Explanation: As Jenny arranges in 4 rows, each row contains 12 tiles. So 12×4= 48. The tiles left are 50-48=2.

Question 4.
Jerome added two numbers. The sum was 83. One of the numbers was 45. What was the other number?
Options:
a. 38
b. 48
c. 42
d. 128

Answer: a

Explanation: The sum of two numbers is 83, in that one number is 45. To find another number we will do subtraction,
i.e 83-45=38.

Go Math Workbook Grade 4 Lesson 5.4 Factors and Multiples Question 5.
There are 18 rows of seats in the auditorium. There are 24 seats in each row. How many seats are in the auditorium in all?
Options:
a. 42
b. 108
c. 412
d. 432

Answer: d

Explanation:
No.of rows= 18, each row has 24 seats. So total no.of seats are 18×24= 432.

Question 6.
The population of Riverdale is 6,735. What is the value of the 7 in the number 6,735?
Options:
a. 7
b. 700
c. 735
d. 7,000

Answer: b

Explanation: In 6,735 the 7 is in the Hundreds Place. So the answer is 7.

Page No. 307

Question 1.
Use the grid to model the factors of 18. Tell whether 18 is prime or composite.

Factors of 18: ____ , ____ , ____ , ____ , ____ , ____
Think: 18 has more than two factors.
So, 18 is _________ .

Answer: Composite number.

Explanation: The number which has more than two factors is called composite numbers.
Factors of 18 are: 1,2,3,6,9,18.

Tell whether the number is prime or composite.

Question 2.
11
Think: Does 11 have other factors besides 1 and itself?

Answer: Prime number.

Explanation: A Prime number is a number that is divisible 1 and itself.

Question 3.
73

Answer: Prime number

Explanation: A Prime number is a number that is divisible 1 and itself.

Question 4.
69

Answer: Composite number.

Explanation: The number which has more than two factors is called composite numbers.
Factors of 69 are: 1,3,23,69.

Question 5.
42

Answer: Composite number.

Explanation: The number which has more than two factors is called composite numbers.
Factors of 42 are: 1,2,3,6,7,21,42.

Tell whether the number is prime or composite.

Question 6.
18

Answer: Composite number.

Explanation: The number which has more than two factors is called composite numbers.
Factors of 18 are: 1,2,3,6,9,18.

Question 7.
49

Answer: Composite number.

Explanation: A number which has more than two factors is called composite numbers.
Factors of 49 are 1,7,49.

Question 8.
29

Answer: Prime number.

Explanation: A Prime number is a number that is divisible 1 and itself.

Question 9.
64

Answer: Composite number.

Explanation: The number which has more than two factors is called composite numbers.
Factors of 64 are: 1,2,4,8,32,64.

Question 10.
33

Answer: Composite number.

Explanation: The number which has more than two factors is called composite numbers.
Factors of 33 are: 1,3,11,33.

Question 11.
89

Answer: Prime number.

Explanation: A Prime number is a number that is divisible 1 and itself.

Question 12.
52

Answer: Composite number.

Explanation: The number which has more than two factors is called composite numbers.
Factors of 52 are: 1,2,4,13,26,52.

Question 13.
76

Answer: Composite number.

Explanation: The number which has more than two factors is called composite numbers.
Factors of 76 are: 1,2,4,19,38,76.

Write true or false for each statement. Explain or give an example to support your answer.

Question 14.
Only odd numbers are prime numbers.

Answer: False.

Explanation: Not all odd numbers are prime numbers. For example. 39 is an odd number but not a prime number because it is divisible by 3 and 13.

Question 15.
A composite number cannot have three factors.

Answer: False

Explanation: A Composite number is a number that has more than two factors.
For example. 21 is a composite number and the factors of 21 are 1,3,7,21.

Question 16.
I am a number between 60 and 100. My ones digit is two less than my tens digit. I am a prime number. What number am I?

Answer: 97

Explanation:
Prime numbers between 60 to 100 are 61,67,71,73,79,83,89,97. 97 is the number which ones digit is two less than tens digit.

Question 17.
Name a 2-digit odd number that is prime. Name a 2-digit odd number that is composite.

Answer:
2 digit Prime odd numbers are 11,13,17 etc.
2 digit Composite odd numbers are 15,21,39

Explanation: A Prime number is a number that is divisible 1 and itself.
The number which has more than two factors is called composite numbers.

Question 18.
Choose the words that correctly complete the sentence.
The number 9 is img 18 because it has img 19 two factors.
Type below:
__________

Page No. 308

The Sieve of Eratosthenes

Eratosthenes was a Greek mathematician who lived more than 2,200 years ago. He invented a method of finding prime numbers, which is now called the Sieve of Eratosthenes.

Question 19.
Follow the steps below to circle all prime numbers less than 100. Then list the prime numbers.
STEP 1
Cross out 1, since 1 is not prime.
STEP 2
Circle 2, since it is prime. Cross out all other multiples of 2.
STEP 3
Circle the next number that is not crossed out. This number is prime. Cross out all the multiples of this number.
STEP 4
Repeat Step 3 until every number is either circled or crossed out.

So, the prime numbers less than 100 are

Answer: 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97.

Explanation: A Prime number is a number that is divisible 1 and itself.

Question 20.
Explain why the multiples of any number other than 1 are not prime numbers.

Answer:

Common Core – Prime and Composite Numbers – Page No. 309

Prime and Composite Numbers

Tell whether the number is prime or composite

Question 1.

Question 2.
68

Answer: Composite number.

Explanation: The number which has more than two factors is called composite numbers.
Factors of 68 are: 1,2,4,17,34,69.

Question 3.
52

Answer: Composite number.

Explanation: The number which has more than two factors is called composite numbers.
Factors of 52 are: 1,2,4,13,26,52.

Question 4.
63

Answer: Composite number.

Explanation: The number which has more than two factors is called composite numbers.
Factors of 63 are: 1,2,3,7,9,21,63.

Go Math Grade 4 Lesson 5.5 Answer Key Question 5.
75

Answer: Composite number.

Explanation: The number which has more than two factors is called composite numbers.
Factors of 75 are: 1,3,5,15,25,75

Question 6.
31

Answer: Prime number.

Explanation: 31 is a prime number that means it is divisible by 1 and itself.

Question 7.
77

Answer: Composite number.

Explanation: The number which has more than two factors is called composite numbers.
Factors of 77 are: 1,7,11,77.

Question 8.
59

Answer: Prime number

Explanation: 59 is a prime number that means it is divisible by 1 and itself.

Question 9.
87

Answer: Composite Number.

Explanation: The number which has more than two factors is called composite numbers.
Factors of 87 are: 1,3,29,87.

Question 10.
72

Answer: Composite Number.

Explanation: The number which has more than two factors is called composite numbers.
Factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72.

Question 11.
49

Answer: Composite Number.

Explanation: The number which has more than two factors is called composite numbers.
Factors of 49 are 1,7,49.

Question 12.
73

Answer: Prime number.

Explanation: A Prime number is a number that is divisible 1 and itself.

Problem Solving

Question 13.
Kai wrote the number 85 on the board. Is 85 prime or composite?

Answer: Composite number

Explanation: The number which has more than two factors is called composite numbers.
Factors of 85 are 1,5,17,85

Question 14.
Lisa says that 43 is a 2-digit odd number that is composite. Is she correct?

Answer: No

Explanation: 43 is a prime number. A Prime number is a number that is divisible 1 and itself.

Common Core – Prime and Composite Numbers – Page No. 310

Lesson Check

Question 1.
The number 5 is:
Options:
a. prime
b. composite
c. both prime and composite
d. neither prime nor composite

Answer: Prime number.

Explanation: A Prime number is a number that is divisible 1 and itself.

Question 2.
The number 1 is:
Options:
a. prime
b. composite
c. both prime and composite
d. neither prime nor composite

Answer: d

Explanation: A Prime number is a number that is divisible 1 and itself. So prime number should have two divisors but 1 has only one divisor. The number which has more than two factors is called composite numbers. So 1 doesn’t have more than two factors. So 1 is neither Prime nor Composite.

Spiral Review

Question 3.
A recipe for a vegetable dish contains a total of 924 calories. The dish serves 6 people. How many calories are in each serving?
Options:
a. 134 calories
b. 150 calories
c. 154 calories
d. 231 calories

Answer: c

Explanation: Total no.of calories are 924, which serves 6 people. To find each serving we will perform division
924÷6= 154 calories.

Math Book Answers Grade 4 Lesson 5.5 Prime and Composite Numbers Question 4.
A store clerk has 45 shirts to pack in boxes. Each box holds 6 shirts. What is the fewest boxes the clerk will need to pack all the shirts?
Options:
a. 9
b. 8
c. 7
d. 6

Answer: b

Explanation: As the box holds only 6 shirts, 42 shirts are packed in 7 boxes, and the remaining 3 shirts will be packed in another box. So the total number of boxes is 8.

Question 5.
Which number rounds to 200,000?
Options:
a. 289,005
b. 251,659
c. 152,909
d. 149,889

Answer: c

Explanation: 152,909 is nearest to 200,000.

Question 6.
What is the word form of the number 602,107?
Options:
a. six hundred twenty thousand, seventeen
b. six hundred two thousand, one hundred seven
c. six hundred twenty-one thousand, seventeen
d. six hundred two thousand, one hundred seventy

Answer: b

Page No. 313

Use the rule to write the numbers in the pattern.

Question 1.
Rule: Subtract 10. First term: 100

Answer: 100,90,80,70,60,..

Explanation:
100
100-10= 90
90-10= 80
80-10= 70
70-10= 60

Use the rule to write the numbers in the pattern.
Describe another pattern in the numbers.

Question 2.
Rule: Multiply by 2. First term: 4
4 , _____ , _____ , _____ , _____ , …….

Answer: 4,8,16,32,64,…

Explanation:
4
4×2= 8
8×2= 16
16×2= 32
32×2= 64

Question 3.
Rule: Skip-count by 6. First term: 12
12 , _____ , _____ , _____ , _____ , …….

Answer: 12,18,24,30,36,…

Explanation:
12
12+6= 18
18+6= 24
24+6= 30
30+6= 36

Use the rule to write the first twelve numbers in the pattern. Describe another pattern in the numbers.

Question 4.
Rule: Add 7. First term: 3

Answer:
3
3+7= 10
10+7= 17
17+7= 34
34+7= 41
41+7= 48
48+7= 55
55+7= 62
62+7= 69
69+7= 76
76+7= 83
83+7= 90.

Explanation: Added 7 to the given term.

Question 5.
5. Rule: Add 2, add 1. First term: 12

Answer: 12,14,15,17,19,21,22,24,25,27,28,30,31.

Explanation:
12
12+2= 14
14+1= 15
15+2= 17
17+1= 19
19+2= 21
21+1= 22
22+2= 24
24+1= 25
25+2= 27
27+1= 28
28+2= 30
30+1= 31

Go Math 4th Grade Chapter 5 Review Test Answer Key Question 6.
Use Patterns Marcie likes to collect stickers, but she also likes to give them away. Currently, Marcie has 87 stickers in her collection. If Marcie collects 5 new stickers each week and gives away 3 stickers each week, how many stickers will Marcie have in her collection after 5 weeks?
_______ stickers

Answer: 97 stickers

Explanation: Marcie has 87 stickers, in 1st week she collected 5 stickers and gave away 3 stickers, which means
87+5-3= 89
2nd week 89+5-3= 91
3rd week 91+5-3= 93
4th week 93+5-3= 95
5th week 95+5-3= 97.

Page No. 314

Question 7.
John is saving for his trip to see the Alamo. He started with $24 in his savings account. Every week he earns $15 for baby-sitting. Out of that, he spends $8 and saves the rest. John uses the rule add 7 to find out how much money he has at the end of each week. What are the first 8 numbers in the pattern?

Answer: $24, $31, $38, $45, $52, $59, $66, $73.

Explanation:
24
24+7= 31
31+7= 38
38+7= 45
45+7= 52
52+7= 59
59+7= 66
66+7= 73.

Question 8.
Draw a check under the column that describes the number.

Pose a Problem

Question 9.
An activity at the Math Fair shows two charts.

Use at least two of the numbers and an operation from the charts to write a pattern problem. Include the first five terms of your pattern in the solution to your problem.
Pose a problem. Solve your problem.
Describe other patterns in the terms you wrote.

Answer:
2+3= 5    Addition.
10-6= 4   Subtraction.
5×2= 10  Multiplication.

Common Core – Number Patterns – Page No. 315

Number Patterns

Use the rule to write the first twelve numbers in the pattern.

Describe another pattern in the numbers.

Question 1.
Rule: Add 8. First term: 5

Question 2.
Rule: Subtract 7. First term: 95

Answer: 95,88,81,74,67,60,53,46,39,32,25,118,11.

Explanation: 95
95-7= 88
88-7= 81
81-7= 74
74-7= 67
67-7= 60
60-7= 53
53-7= 46
46-7= 39
39-7= 32
32-7= 25
25-7= 18
18-7= 11

Go Math Grade 4 Teacher Edition Answers Pdf Chapter 5 Question 3.
Rule: Add 15, subtract 10. First term: 4

Answer: 4,19,9,24,14,29,19,34,24,39,29,44,34.

Explanation: 4
4+15= 19
19-10= 9
9+15= 24
24-10= 14
14+15= 29
29-10= 19
19+15= 34
34-10= 24
24+15= 39
39-10=29
29+15=44
44-10=34

Question 4.
Rule: Add 1, multiply by 2. First term: 2

Answer: 2,4,5,10,11,22,23,46,47,94,95,190.

Explanation: 2
2+1= 2
2×2= 4
4+1= 5
5×2= 10
10+1= 11
11×2= 22
22+1= 23
23×2= 46
46+1= 47
47×2= 94
94+1= 95
95×2= 190.

Problem Solving

Question 5.
Barb is making a bead necklace. She strings 1 white bead, then 3 blue beads, then 1 white bead, and so on. Write the numbers for the first eight beads that are white. What is the rule for the pattern?

Answer:

Explanation: 1
1+4= 5
5+4= 9
9+4= 13
13+4= 17
17+4= 21
21+4= 25
25+4=29

Question 6.
An artist is arranging tiles in rows to decorate a wall. Each new row has 2 fewer tiles than the row below it. If the first row has 23 tiles, how many tiles will be in the seventh row?

Answer: 11 tiles.

Explanation: 23
23-2= 21
21-2= 19
19-2= 17
17-2= 15
15-2= 13
13-2= 11

Common Core – Number Patterns – Page No. 316

Lesson Check

Question 1.
The rule for a pattern is add 6. The first term is 5. Which of the following numbers is a term in the pattern?
Options:
a. 6
b. 12
c. 17
d. 22

Answer: c

Explanation: 5
5+6= 11
11+6= 17

Question 2.
What are the next two terms in the pattern 3, 6, 5, 10, 9, 18, 17, . . .?
Options:
a. 16, 15
b. 30, 31
c. 33, 34
d. 34, 33

Answer: d

Explanation: 3
3×2= 6
6-1= 5
5×2= 10
10-1= 9
9×2= 18
18-1= 17
17×2= 34
34-1= 33

Spiral Review

Question 3.
To win a game, Roger needs to score 2,000 points. So far, he has scored 837 points. How many more points does Roger need to score?
Options:
a. 1,163 points
b. 1,173 points
c. 1,237 points
d. 2,837 points

Answer: a

Explanation: Roger has scored 837 points, He needs to score 2000 points to win, So to know how much more points do Roger needs we need to subtract i.e 2,000-837= 1,163.

Question 4.
Sue wants to use mental math to find 7 × 53. Which expression could she use?
Options:
a. (7 × 5) + 3
b. (7 × 5) + (7 × 3)
c. (7 × 50) + 3
d. (7 × 50) + (7 × 3)

Answer: d

Explanation: Distributive property means if we multiply a sum by a number is the same as multiplying each addend by the number and adding the products.

Chapter 5 Review Test Go Math 4th Grade Answer Key Question 5.
Pat listed numbers that all have 15 as a multiple. Which of the following could be Pat’s list?
Options:
a. 1, 3, 5, 15
b. 1, 5, 10, 15
c. 1, 15, 30, 45
d. 15, 115, 215

Answer: a

Explanation:
1×15= 15
3×5= 15
5×3= 15
15×1= 15

Question 6.
Which is a true statement about 7 and 14?
Options:
a. 7 is a multiple of 14.
b. 14 is a factor of 7.
c. 14 is a common multiple of 7 and 14.
d. 21 is a common multiple of 7 and 14.

Answer: c

Explanation:
7×2=14
14×1=14

Review/Test – Page No. 317

Question 1.
List all the factors of the number.
14: ______ , ______ , ______ , ______

Answer: 1,2,7,14

Explanation: Factors are the numbers that divide the original number completely. Here, we can see the numbers which give the result as 14 when multiplied together. So the factors of 14 are 1,2,7,14.

Question 2.
Select the numbers that have a factor of 5. Mark all that apply.
Options:
a. 15
b. 3
c. 45
d. 5
e. 50
f. 31

Answer: a,c,d,e.

Explanation: Factors are the numbers that divide the original number completely.

Question 3.
Jackson was making a poster for his room. He arranged 50 trading cards in the shape of a rectangle on the poster.
For 3a–3e, choose Yes or No to tell whether a possible arrangement of cards is shown.
a. 5 rows of 10 cards
i. yes
ii. no

Answer: Yes

Explanation: 5 rows of 10 cards that means 5×10= 50. So the answer is Yes.

Question 3.
b. 7 rows of 8 cards
i. yes
ii. no

Answer: No

Explanation: 7×8= 56, There will be extra cards. So the answer is No.

Question 3.
c. 25 rows of 2 cards
i. yes
ii. no

Answer: Yes.

Explanation: 25×2=50. So the answer is Yes

Question 3.
d. 50 rows of 1 card
i. yes
ii. no

Answer: Yes

Explanation: 50×1=50. So the answer is Yes.

Question 3.
e. 45 rows of 5 cards
i. yes
ii. no

Answer: No

Explanation: 45×5= 225. Which is not equal to 50. So the answer is No.

Go Math Grade 4 Lesson 5.6 Answer Key Question 4.
List all the factor pairs in the table.

Answer:
1×48= 48   1,48
2×24= 48   2,24
3×16= 48   3,16
4×12= 48   4,12
6×8=  48    6,8

Explanation: Factors are the numbers that divide the original number completely. Here, we can see the numbers which give the result as 30 when multiplied together.

Review/Test – Page No. 318

Question 5.
Classify the numbers. Some numbers may belong in more than one box.

Answer:
Divisible by 5 and 9: 90
Divisible by 6 and 9: 54,72,90
Divisible by 2 and 6: 54,72,84,90,96

Question 6.
James works in a flower shop. He will put 36 tulips in vases for a wedding. He must use the same number of tulips in each vase. The number of tulips in each vase must be greater than 1 and less than 10. How many tulips could be in each vase?

Answer: 2, 3, 4, 6, 9.

Explanation:

Question 7.
Brady has a card collection with 64 basketball cards, 32 football cards, and 24 baseball cards. He wants to arrange the cards in equal piles, with only one type of card in each pile. How many cards can he put in each pile? Mark all that apply.
Options:
a. 1
b. 2
c. 3
d. 4
e. 8
f. 32

Answer: a,b,d,e

Explanation:
Factors of 64 are 1,2,4,8,16,32,64.
Factors of 32 are 1,2,4,8,16,32.
Factors of 24 are 1,2,3,4,6,8,12,24.
Common factors are 1,2,4,8.

Go Math Grade 4 Chapter 5 Test Answers Question 8.
The Garden Club is designing a garden with 24 cosmos, 32 pansies, and 36 marigolds. Each row will have only one type of plant in each row. Ben says he can put 6 plants in each row. He listed the common factors of 24, 32, and 36 below to support his reasoning.
24: 1, 2, 3, 4, 6, 8, 12, 24
32: 1, 2, 4, 6, 9, 16, 32
36: 1, 2, 3, 4, 6, 8, 12, 18, 36
Is he correct? Explain your answer. If his reasoning is incorrect, explain how he should have found the answer.

Answer: No. He can put 1,2,4 plants in each row

Explanation: The factors of 32 are incorrect. He listed as 6 and 9 are factors of 32 which is wrong and 8 is not a factor of 36.
Factors of 32 are 1,2,4,8,16,32.
Factors of 36 are 1,2,3,4,6,9,18,36.
Common factors of 24,32 and 36 are 1,2,4. So he can put 1,2,4 plants in each row.

Review/Test – Page No. 319

Question 9.

Part A
The museum is hosting a show in July that features oil paintings by different artists. All artists show the same number of paintings and each will show more than 1 painting. How many artists could be featured in the show?

Answer: 2,3,5,6,10,15

Explanation:
Factors of 30 are 1,2,3,5,6,10,15,30.

Question 9.
Part B
The museum wants to display all the art pieces in rows. Each row has the same number of pieces and the same type of pieces. How many pieces could be in each row? Explain how you found your answer.

Answer: 1,3.

Explanation:
Factors of 30 are 1,2,3,5,6,10,15,30.
Factors of 24 are 1,2,3,4,6,8,12,24
Factors of 21 are 1,3,7,21
Common Factors are 1,3

4th Grade Go Math Chapter 5 Review Test Question 10.
Charles was skip counting at the Math Club meeting. He started to count by 8s. He said 8, 16, 24, 32, 40, and 48. What number will he say next?

Answer: 56

Explanation: Multiples of 8
8×1= 8
8×2= 16
8×3= 24
8×4= 32
8×5= 40
8×6= 48
8×7= 56.

Question 11.
Jill wrote the number 40. If her rule is to add 7, what is the fourth number in Jill’s pattern? How can you check your answer?

Answer: 61

Explanation:
40
40+7= 47
47+7= 54
54+7= 61, And the fourth number is 61

Review/Test – Page No. 320

Question 12.
For numbers 12a–12e, select True or False for each statement.
a. The number 36 is a multiple of 9.
i. True
ii. False

Answer: True

Explanation: 9×4= 36.

Question 12.
b. The number 3 is a multiple of 9.
i. True
ii. False

Answer: False

Explanation: Multiples of 9 are 9,18,27,36,45,54,63, etc.

Question 12.
c. The number 54 is a multiple of 9.
i. True
ii. False

Answer: True

Explanation: 9×6= 54

Question 12.
d. The number 3 is a factor of 9.
i. True
ii. False

Answer: True

Explanation: Factors of 9 are 1,3,9.

Question 12.
e. The number 27 is a factor of 9.
i. True
ii. False

Answer: True

Explanation: Factors of 27 are 1,3,9,27

Question 13.
What multiple of 7 is also a factor of 7?

Answer: 7

Explanation: 7 is both multiple and a factor of 7.

Go Math Grade 4 Assessment Guide Pdf Question 14.
Manny makes dinner using 1 box of pasta and 1 jar of sauce. If pasta is sold in packages of 6 boxes and sauce is sold in packages of 3 jars, what is the least number of dinners that Manny can make without any supplies leftover?

Answer: 6

Manny has 1 box of pasta and 1 jar of sauce and he sold in a package of 6 boxes of pasta and 3 jars of sauce. Let the packages of pasta be 6P and jars of sauce be 3s.
As Manny sold without any leftover 3S=6P,
If we take 1 package of pasta then P=1,
And 3S=6×1, where S= 6/3 which is equal to 2,
So for every package of pasta, we need 2 packages of sauce,
So the minimum purchase is 2 packages of sauce and 1 package of pasta. Since pasta packages are 6 boxes the minimum number of meals is 6.

Question 15.
Serena has several packages of raisins. Each package contains 3 boxes of raisins. What could be the number of boxes of raisins Serena has? Mark all that apply.
Options:
a. 9
b. 18
c. 23
d. 27
e. 32

Answer: a,b,d

Explanation: Factors of 3.

Question 16.
Choose the words that make the sentence true.
The number 7 is because it has two factors.
The number 7 is _________ because it has
_________ two factors.

Answer: The number 7 is a prime number because it has exactly two factors.

Explanation: A Prime number is a number that is divisible 1 and itself.

Review/Test – Page No. 321

Question 17.
Winnie wrote the following riddle: I am a number between 60 and 100. My ones digit is two less than my tens digit. I am a prime number.
Part A
What number does Winnie’s riddle describe? Explain.

Answer: 97

Explanation: 97 is the number which ones digit is two less than tens digit.

Question 17.
Part B
Winnie’s friend Marco guessed that her riddle was about the number 79. Why can’t 79 be the answer to Winnie’s riddle?
Explain.

Answer: It’s wrong because in Winnie’s riddle ones digit is two less than tens digit. But in 79 ones digit is two greater than tens digit.

Explanation: In 79 ones digit is two greater than tens digit. So Marco guess was incorrect.

Question 18.
Classify the numbers as prime or composite.

Answer: Prime numbers are 37, 71
Composite numbers are 65, 82

Explanation:
A Composite number is a number that has more than two factors.
A Prime number is a number that is divisible 1 and itself.

Question 19.
Erica knits 18 squares on Monday. She knits 7 more squares each day from Tuesday through Thursday. How many squares does Erica knit on Friday?

Answer: 46 squares.

Explanation: 18
18+7= 25
25+7= 32
32+7= 39
39+7= 46.

Question 20.
Use the rule to write the first five terms of the pattern.
Rule: Add 10, subtract 5
First term: 11 ______ ______ ______ ______

Answer: 11,21,16,26,21.

Explanation: 11
11+10= 21
21-5= 16
16+10= 26
26-5= 21

Review/Test – Page No. 322

Question 21.
Elina had 10 tiles to arrange in a rectangular design. She drew a model of the rectangles she could make with the ten tiles.

Part A
How does Elina’s drawing show that the number 10 is a composite number?

Answer: 10 is a composite number because it has more than two factors.

Explanation: A number which has more than two factors is called a composite numbers.

Question 21.
Part B
Suppose Elina used 15 tiles to make the rectangular design. How many different rectangles could she make with the 15 tiles? Write a list or draw a picture to show the number and dimensions of the rectangles she could make.

Answer: 2

Explanation: one by 15 tiles and second by 3tiles in a row.

Question 21.
Part Cs
Elina’s friend Luke said that he could make more rectangles with 24 tiles than with Elina’s 10 tiles. Do you agree with Luke? Explain.

Answer: Yes

Explanation: As 24 has more factors than 10.

Page No. 329

Use the model to write an equivalent fraction.

Question 1.


\(\frac{1}{5}\) = \(\frac{□}{□}\)

Answer: 1/5= 2/10

Explanation: From the above figure we can see that there are 5 equal parts and in that 1 part is shaded. So the fraction of the shaded part is 1/5.

Question 2.


\(\frac{2}{3}\) = \(\frac{□}{□}\)

Answer: 2/3= 6/9

Explanation: From the above figure we can see that there are 3 equal parts and in that 2 part is shaded. So the fraction of the shaded part is 2/3.

Tell whether the fractions are equivalent. Write = or ≠.

Question 3.
\(\frac{1}{6}\) _____ \(\frac{2}{12}\)

Answer: 1/6=2/12

Explanation: The denominator and numerators are equal for both fractions. So 1/6=2/12 are equal.

Question 4.
\(\frac{2}{5}\) _____ \(\frac{6}{10}\)

Answer: 2/5≠ 6/10

Explanation: The denominator and numerators are not equal for both the fractions.

Question 5.
\(\frac{4}{12}\) _____ \(\frac{1}{3}\)

Answer: 4/12=1/3

Explanation: The denominator and numerators are equal for both fractions.

Question 6.
\(\frac{5}{8}\) _____ \(\frac{2}{4}\)

Answer: 5/8≠2/4

Explanation: The denominator and numerators are not equal for both the fractions.

Question 7.
\(\frac{5}{6}\) _____ \(\frac{10}{12}\)

Answer: 5/6=10/12

Explanation: The denominator and numerators are equal for both fractions.

Go Math Grade 4 Answer Key Pdf Question 8.
\(\frac{1}{2}\) _____ \(\frac{5}{10}\)

Answer: 1/2=5/10

Explanation: The denominator and numerators are equal for both fractions.

Question 9.
Manny used 8 tenth-size parts to model \(\frac{8}{10}\). Ana used fewer parts to model an equivalent fraction. How does the size of a part in Ana’s model compare to the size of a tenth-size part? What size part did Ana use?

Answer: Larger than a tenth-size part. And she used the fifth-size part.

Explanation: A part of Ana’s model is larger than a tenth-size part. And she used the fifth-size part.

Question 10.
Use a Concrete Model How many eighth-size parts do you need to model \(\frac{3}{4}\)? Explain.

Answer: 6

Explanation: Let the parts be X, then 1/8×X=3/4. By calculation, we will get X as 6.
So we need 6 parts.

Page No. 330

Question 11.
Ben brought two pizzas to a party. He says that since 14_ of each pizza is left, the same amount of each pizza is left. What is his error?

Answer: As the size of pizzas is not the same, 1/4 of leftover pizza is not equal to another.

Go Math Grade 4 Chapter 5 Answer Key Pdf Question 12.
For numbers 12a–12d, tell whether the fractions are equivalent by selecting the correct symbol.
a. \(\frac{3}{15}\) _____ \(\frac{1}{6}\)

Answer: 3/5≠1/6

Question 12.
b. \(\frac{3}{4}\) _____ \(\frac{16}{20}\)

Answer: 3/4≠16/20

Question 12.
c. \(\frac{2}{3}\) _____ \(\frac{8}{12}\)

Answer: 2/3=8/12

Question 12.
d. \(\frac{4}{5}\) _____ \(\frac{8}{10}\)

Answer: 4/5=8/10.

Go Math Grade 7 Answer Key Chapter 12 Experimental Probability contains guided practice which helps students to prepare for their exams and can score the highest marks. All students of 7th grade can refer and Download Go Math Answer Key Chapter 12 Experimental Probability. We have provided a brief explanation which makes students to understand easily. By this Go Math Grade 7 Answer Key Chapter 12 Experimental Probability students can score good marks in the exams.

Go Math Grade 7 Answer Key Chapter 12 Experimental Probability

In this chapter, every question was explained step by step by that students will not face any difficulty while preparing for exams. Every chapter was explained in an easy way so that students can understand easily. Click the below links for the preparation.

Chapter 12– Lesson 1:

• Guided Practice – Page No. 372
• Independent Practice – Page No. 373
• Page No. 374

Chapter 12– Lesson 2:

• Guided Practice – Page No. 378
• Independent Practice – Page No. 379
• Page No. 380

Chapter 12– Lesson 3:

• Guided Practice – Page No. 384
• Independent Practice – Page No. 385
• H.O.T. – Page No. 386

Chapter 12– Lesson 4:

• Guided Practice – Page No. 390
• Independent Practice – Page No. 391
• Page No. 392
• 12.1 Probability – Page No. 393
• Selected Response – Page No. 394

Guided Practice – Page No. 372

Determine whether each event is impossible, unlikely, as likely as not, likely, or certain. Then, tell whether the probability is 0, close to 0, \(\frac{1}{2}\) , close to 1, or 1.

Question 2.
randomly picking a green card from a standard deck of playing cards
_____

Answer: Probability is 0.

Explanation: A standard deck of play cards does not have green cards, so the probability is 0.

Question 3.
randomly picking a red card from a standard deck of playing cards
_____

Answer: Probability is 1/2.

Explanation: As half of the cards are from a standard deck of playing cards are red, so the probability is 1/2.

Question 4.
picking a number less than 15 from a jar with papers labeled from 1 to 12
_____

Answer: The probability of picking a number less than 15 is 1.

Explanation: All papers have a number less than 15, so the probability of picking a number less than 15 is 1.

Lesson 12.1 Probability Answers Question 5.
picking a number that is divisible by 5 from a jar with papers labeled from 1 to 12
close to _____

Answer: The probability is 1/6.

Explanation: The numbers divisible by 5 from 1 to 12 are 5 and 10, so there are two positive numbers from a total of 12 positive numbers and the probability of picking a number that is divisible by 5 is 2/12= 1/6.

Find each probability. Write your answer in the simplest form.

Question 6.
spinning a spinner that has 5 equal sections marked 1 through 5 and landing on an even number
\(\frac{□}{□}\)

Answer: The spinner landing on an even number is 2/5.

Explanation: There are 5 possible outcomes when spinning the spinner and there are two even numbers on the spinner 2 and 4. So the probability of the spinner landing on an even number is 2/5.

Question 7.
picking a diamond from a standard deck of playing cards which has 13 cards in each of four suits: spades, hearts, diamonds, and clubs
\(\frac{□}{□}\)

Answer: The probability is 1/4.

Explanation: As the deck of cards contains 52 cards. so there are 52 possible outcomes in the situation and there are 13 cards with diamonds. So the probability is 13/52= 1/4.

Use the complement to find each probability.

Question 8.
What is the probability of not rolling a 5 on a standard number cube?
\(\frac{□}{□}\)

Answer: The probability of not rolling a 5 is 5/6.

Explanation: A standard number cube has six sides and marked from 1 to 6, so the probability of rolling a 5 is 1/6. And the probability of not rolling a 5 is
P(rolling a 5) + P(not rolling a 5 ) = 1
1/6 + P(not rolling a 5 )= 1
P(not rolling a 5 ) = 1- 1/6
= (6-1)/6
= 5/6.

Question 9.
A spinner has 3 equal sections that are red, white, and blue. What is the probability of not landing on blue?
\(\frac{□}{□}\)

Answer: The probability of not landing on blue is 2/3.

Explanation: As spinner has 3 equal sections, so possible outcomes are 3. The probability of landing on blue is 1/3, so the probability of not landing on blue is
P(landing on blue) + P(not landing on blue ) = 1
1/3 + P(not landing on blue )= 1
P(not landing on blue ) = 1- 1/3
= (3-1)/3
= 2/3.

Question 10.
A spinner has 5 equal sections marked 1 through 5. What is the probability of not landing on 4?
\(\frac{□}{□}\)

Answer:

Explanation: As spinner has 5 equal sections, so possible outcomes are 5. The probability of landing on 4 is 1/5, so the probability of not landing on 4 is
P(landing on 4) + P(not landing on 4 ) = 1
1/5 + P(not landing on blue )= 1
P(not landing on blue ) = 1- 1/5
= (5-1)/5
= 4/5.

Probability Answer Key Question 11.
There are 4 queens in a standard deck of 52 cards. You pick one card at random. What is the probability of not picking a queen?
\(\frac{□}{□}\)

Answer: The probability of not picking a queen is 12/13.

Explanation: The deck has 52 cards, so there are 52 possible outcomes. And there are 4 queens in the deck, so the probability of picking a queen is 4/52= 1/13. And the probability of not picking a queen is
P(picking queen) + P(not picking queen ) = 1
1/13 + P(not picking queen )= 1
P(not picking queen ) = 1- 1/13
= (13-1)/13
= 12/13.

Essential Question Check-In

Question 12.
Describe an event that has a probability of 0% and an event that has a probability of 100%.

Answer: An event with a probability of 0% would be picking a number card labeled 15 from a standard deck of cards. And an event with a probability of 100% would be picking a red marble from a bowl filled only with red marbles.

Independent Practice – Page No. 373

Question 13.
There are 4 aces and 4 kings in a standard deck of 52 cards. You pick one card at random. What is the probability of selecting an ace or a king? Explain your reasoning.
\(\frac{□}{□}\)

Answer: The probability of selecting an ace or king from the deck is 2/13.

Explanation: There are 52 possible outcomes when picking from a deck of cards and there are 8 cards that have an ace or king, so the probability of selecting an ace or king from the deck is 8/52= 2/13.

Question 14.
There are 12 pieces of fruit in a bowl. Seven of the pieces are apples and two are peaches. What is the probability that a randomly selected piece of fruit will not be an apple or a peach? Justify your answer.
\(\frac{□}{□}\)

Answer: The probability of not picking an apple or a peach is 1/4.

Explanation: There are 12 possible outcomes and 9 out of 12 fruits are apples or peaches, so the probability of picking an apple or a peach is 9/12 = 3/4. And the probability of not picking an apple or a peach is
P(picking an apple or a peach ) + P(not picking an apple or a peach ) = 1
3/4 + P(not picking an apple or a peach)= 1
P(not picking an apple or a peach) = 1- 3/4
= (4-3)/4
= 1/4.

Probability Test Answer Key 7th Grade Question 15.
Critique Reasoning
For breakfast, Clarissa can choose from oatmeal, cereal, French toast, or scrambled eggs. She thinks that if she selects a breakfast at random, it is likely that it will be oatmeal. Is she correct? Explain your reasoning.
_____

Answer: Clarissa is unlikely to have oatmeal.

Explanation: As Clarissa has 4 breakfast choices, the probability of choosing oatmeal is 1/4, so it is unlikely that Clarissa will have oatmeal.

Question 16.
Draw Conclusions
A researcher’s garden contains 90 sweet pea plants, which have either white or purple flowers. About 70 of the plants have purple flowers, and about 20 have white flowers. Would you expect that one plant randomly selected from the garden will have purple or white flowers? Explain.
_____

Answer: The one plant randomly selected from the garden will be purple flowers.

Explanation: The probability of selecting a white-flowered plant is 20/90= 2/9 and the probability of selecting a purple-flowered plant is 70/90= 7/9. So, it is more likely to select a purple plant when randomly choosing from the garden

Question 17.
The power goes out as Sandra is trying to get dressed. If she has 4 white T-shirts and 10 colored T-shirts in her drawer, is it likely that she will pick a colored T-shirt in the dark? What is the probability she will pick a colored T-shirt? Explain your answers.
_____

Answer: The probability of selecting a colored T-shirt is 5/7.

Explanation: The total number of T-shirts in Sandra’s drawer is 14, and the probability of selecting a white T-shirt is 4/14= 2/7. And the probability of selecting a colored T-shirt is 10/14= 5/7. So, it is more likely to choose a colored T-shirt when randomly choosing from the drawer.

Page No. 374

Question 18.
James counts the hair colors of the 22 people in his class, including himself. He finds that there are 4 people with blonde hair, 8 people with brown hair, and 10 people with black hair. What is the probability that a randomly chosen student in the class does not have red hair? Explain.
_____

Answer: The probability that a randomly chosen student in the class does not have red hair is 1.

Explanation: As there is no student with red hair in James’ class and that means the probability that a randomly chosen student in the class does not have red hair is 1.

Question 19.
Persevere in Problem-Solving
A bag contains 8 blue coins and 6 red coins. A coin is removed at random and replaced by three of the other colors.
a. What is the probability that the removed coin is blue?
\(\frac{□}{□}\)

Answer: The probability that the blue coin is removed is 8/14= 4/7.

Explanation: The total number of coins in the bag is 14, so there are 14 possible outcomes. As there are 8 blue coins and 6 red coins, so the probability that the blue coin is removed is 8/14= 4/7.

Question 19.
b. If the coin removed is blue, what is the probability of drawing a red coin after three red coins are put in the bag to replace the blue one?
\(\frac{□}{□}\)

Answer: The probability of choosing a red coin is 9/16.

Explanation: The total number of coins in the bag is 14 and one blue coin was removed which means 8-1= 7 and three red coins are added, so 6+3= 9  the total number of coins is 7+9= 16. And there are 7 blue coins and 9 red coins, so the probability of choosing a red coin is 9/16.

Question 19.
c. If the coin removed is red, what is the probability of drawing a red coin after three blue coins are put in the bag to replace the red one?
\(\frac{□}{□}\)

Answer: The probability of choosing a red coin is 5/16.

Explanation: The total number of coins in the bag is 14 and one red coin was removed which means 6-1= 5 and three blue coins are added, so 8+3= 11 the total number of coins is 5+11= 16. And there are 11 blue coins and 5 red coins, so the probability of choosing a red coin is 5/16.

H.O.T.

Focus on Higher Order Thinking

Question 20.
Draw Conclusions
Give an example of an event in which all of the outcomes are not equally likely. Explain.

Answer: A bag of coins with 5 red coins and 11 blue coins are not equally likely.

Explanation: All the outcomes are not equally likely would be having a bag of coins with 5 red coins and 11 blue coins. Since these are not the same number of coins of each color, so the probability of choosing a certain color is not equal.

Question 21.
Critique Reasoning
A box contains 150 black pens and 50 red pens. Jose said the sum of the probability that a randomly selected pen will not be black and the probability that the pen will not be red is 1. Explain whether you agree.

Answer: Jose is correct and the probability of choosing a pen that is not black and the probability of choosing a pen that is not red is 1.

Explanation: Since the pens are either red or black, the probability of choosing a pen that is not black is equal to the probability of choosing a pen that is red and the probability of choosing a pen that is not red is equal to the probability of choosing a pen that is black. So the probability of choosing a pen that is not black and the probability of choosing a pen that is not red is
P(not red)+P(not black)= 150/200 + 50/200
= 200/200
= 1.

Experimental Probability Grade 7 Question 22.
Communicate Mathematical Ideas
A spinner has 7 identical sections. Two sections are blue, 1 is red, and 4 of the sections are green. Suppose the probability of an event happening is \(\frac{2}{7}\). What does each number in the ratio represent? What outcome matches this probability?

Answer: The probability outcome match is 2/7.

Explanation: The spinner has 7 identical sections. The numerator of the ratio represents the number of sections with color and the denominator represents the total number of sections on the spinner. And there are 2 blue sections, so the probability of the spinner landing on the blue is 2/7 matches.

Guided Practice – Page No. 378

Question 1.
A spinner has four sections lettered A, B, C, and D. The table shows the results of several spins. Find the experimental probability of spinning each letter as a fraction in simplest form, a decimal, and a percent.

Answer:
The probability of letter A is 35%.
The probability of letter B is 17.5%.
The probability of letter C is 27.5%.
The probability of letter D is 20%.

Explanation:
The total number of spins is 14+7+11+8=40.
The probability for the letter A is 14/40= 7/20
= 0.35
= 35%.
The probability for the letter B is 7/40= 0.175
= 17.5%.
The probability for the letter C  is 11/40= 0.275
= 27.5%.
The probability for the letter D is 8/40= 1/5
= 0.2
= 20%.

Question 2.
Rachel’s free-throw average for basketball is 60%. She wants to predict how many times in the next 50 tries she will make a free throw. Describe how she could use 10 index cards to predict the answer.

Answer: As Rachel has a 60%  success rate, she could write Successful on 6 cards and Unsuccessful on 4 cards. She then needs to draw cards at random 50 times and record the number of times she gets a Successful card.

Essential Question Check-In

Question 3.
Essential Question Follow-Up
How do you find an experimental probability of a simple event?

Answer: To find the experimental probability of a simple event, divide the number of successful outcomes by the total number of outcomes in the experiment.

Explanation: To find the experimental probability of a simple event, divide the number of successful outcomes by the total number of outcomes in the experiment. For example, if a person makes 10 free throws out of 18, attempts, the experimental probability of making the next free throw is 10/18 = 5/9.

Independent Practice – Page No. 379

Question 4.
Dree rolls a strike in 6 out of the 10 frames of bowling. What is the experimental probability that Dree will roll a strike in the first frame of the next game? Explain why a number cube would not be a good way to simulate this situation.
\(\frac{□}{□}\)

Answer: The experimental probability is 6/10 = 3/5. The number of possible outcomes should be a multiple of 5.

Experiment:
The experimental probability is 6/10 = 3/5. As the denominator of the probability is either 5 or 10, a number cube would not be able to represent the outcomes because it has 6 faces.

Question 5.
To play a game, you spin a spinner like the one shown. You win if the arrow lands in one of the areas marked “WIN”. Lee played this game many times and recorded her results. She won 8 times and lost 40 times. Use Lee’s data to explain how to find the experimental probability of winning this game.

\(\frac{□}{□}\)

Answer: The experimental probability is 8/48= 1/6.

Explanation: As Lee won 8 times and lost 40 times, the number of spins that Lee played is 8+40=48. So the experimental probability is 8/48= 1/6.

Question 6.
The names of the students in Mr. Hayes’ math class are written on the board. Mr. Hayes writes each name on an index card and shuffles the cards. Each day he randomly draws a card, and the chosen student explains a math problem at the board. What is the probability that Ryan is chosen today? What is the probability that Ryan is not chosen today?

Chosen: \(\frac{□}{□}\)

Answer: The probability of the teacher not choosing Ryan is 19/20.

Explanation: As there are 20 students, so possible outcomes are 20. And Ryan is 1 student, the probability of the teacher choosing Ryan is 1/20. And the probability of the teacher not choosing Ryan is
= 1- 1/20
= (20-1)/20
= 19/20.

Question 7.
Critique Reasoning
A meteorologist reports an 80% chance of precipitation. Is this an example of experimental probability, written as a percent? Explain your reasoning.
______

Answer: Yes, the given example is the experimental probability.

Explanation: Experimental probability uses past data to predict future data. The probability that it will rain is based on historical data. So it is an experimental probability written as a percent.

Page No. 380

Question 8.
Mica and Joan are on the same softball team. Mica got 8 hits out of 48 times at bat, while Joan got 12 hits out of 40 times at bat. Who do you think is more likely to get a hit her next time at bat? Explain.
______

Answer: Joan is more likely to get hit her next time at the bat.

Explanation:
As Mica got 8 hits out of 48 times, so the experimental probability of getting a hit is 8/48= 1/6.
And Joan got 12 hits out of 40 times, the experimental probability of getting a hit is 12/40= 3/10.
Therefore Joan is more likely to get hit her next time at the bat.

Experimental Probability 7th Grade Question 9.
Make a Prediction
In tennis, Gabby serves an ace, a ball that can’t be returned, 4 out of the 10 times she serves. What is the experimental probability that Gabby will serve an ace in the first match of the next game? Make a prediction about how many aces Gabby will have for the next 40 servings. Justify your reasoning.
\(\frac{□}{□}\)

Answer: The experimental probability of her serving an ace is 2/5. In 40 serves, she will serve an ace about 2/5×40 = 16 times.

Explanation: To find the experimental probability we need to divide the number of tries by the number of aces. As Gabby serves 4 aces out of 10 times, the experimental probability of her serving an ace is 4/10= 2/5. Next, to make a prediction about how many aces Gabby will have for the next 40 serves, we need to multiply the number of servers by 40 by the experimental probability. In her next 40 serves, she will serve an ace about 2/5×40 = 16 times.

Question 10.
Represent Real-World Problems
Patricia finds that the experimental probability that her dog will want to go outside between 4 P.M. and 5 P.M. is \(\frac{7}{12}\). About what percent of the time does her dog not want to go out between 4 P.M. and 5 P.M.?
______ %

Answer: 41.67%

Explanation: As the sum of the probabilities of an event and its complement is always equal to 1 and P(dog want to go outside) is \(\frac{7}{12}\).
So P(dog want to go outside)+P(dog does not want to go outside) = 1
\(\frac{7}{12}\) + P(dog does not want to go outside) = 1
P(dog does not want to go outside) = 1-\(\frac{7}{12}\)
= \(\frac{12-7}{12}\)
= \(\frac{5}{12}\)
= 0.4167
= 41.67%

H.O.T.

Focus on Higher Order Thinking

Question 11.
Explain the Error
Talia tossed a penny many times. She got 40 heads and 60 tails. She said the experimental probability of getting heads was \(\frac{40}{60}\). Explain and correct her error.

Answer: Talia is not correct.

Explanation: As Taila got 40 heads and 60 tails, which means that she did 100 tosses of the coin. So the experimental probability of getting heads was \(\frac{40}{100}\)

Question 12.
Communicate Mathematical Ideas
A high school has 438 students, with about the same number of males as females. Describe a simulation to predict how many of the first 50 students who leave school at the end of the day are female.

Answer: Since high school has about the same number of male students as female students, the probability of a student leaving school at the end of the day being female is about 50%. And a possible simulation could be using a coin toss, with heads representing males and tails representing females. Toss the coin 50 times and use the results to make a prediction.

Question 13.
Critical Thinking
For a scavenger hunt, Chessa put one coin in each of 10 small boxes. Four coins are quarters, 4 are dimes, and 2 are nickels. How could you simulate choosing one box at random? Would you use the same simulation if you planned to put these coins in your pocket and choose one? Explain your reasoning.

Answer: A possible simulation could be using 10 index cards. Four of the cards could be labeled as quarters, four as dimes, and two nickels. Then cards can be drawn and recorded to simulate choosing a box at random. This simulation could not be used if you planned to put these coins in your pocket and choose one. This is because the size of the coins vary. As we would be able to tell what coin it was in the pocket by feeling its size. And picking one of out your pocket is different than picking a box out of 10 boxes of the same size.

Guided Practice – Page No. 384

Question 1.
A dentist has 400 male and female patients that range in ages from 10 years old to 50 years old and up as shown in the table. What is the experimental probability that the next patient will be female and in the age range 22–39?

\(\frac{□}{□}\)

Answer: The experimental probability is \(\frac{1}{8}\)

Explanation: The total male and female patients are 400.
The age range 22-39 is 50 females
The experimental probability is \(\frac{50}{400}\)
= \(\frac{1}{8}\).

Probability Unit Test Grade 7 Answer Key Question 2.
At a car wash, customers can choose the type of wash and whether to use the interior vacuum. Customers are equally likely to choose each type of wash and whether to use the vacuum. Use a simulation to find the experimental probability that the next customer purchases a deluxe wash and no interior vacuum. Describe your simulation.

Answer: The experimental probability is \(\frac{11}{50}\).

Explanation: A possible simulation could be using a standard cube and flipping a coin. If the number cube rolls 1 or 2 it is recorded as a standard wash, if the number cube rolls 3 or 4 it is recorded as a deluxe wash, if the number cube rolls 5 or 6 it is recorded as a superior wash. For the coin toss, heads count as vacuum and tails count as no vacuum.
For example:

So the experimental probability that the next customer purchases a deluxe and no interior vacuum is \(\frac{11}{50}\).

Essential Question Check-In

Question 3.
How do you find the experimental probability of a compound event?

Answer: To find the experimental probability of a compound event, determine the number of occurrences that satisfy both events and then divide it by the total number of trials.

Independent Practice – Page No. 385

Question 4.
Represent Real-World Problems
For the same food trailer mentioned in Example 1, explain how to find the experimental probability that the next order is two pieces of chicken with a green salad.

\(\frac{□}{□}\)

Answer: The experimental probability is \(\frac{1}{10}\).

Explanation: The total number of orders is 330 and in that 33 orders are with 2 pieces of green salad, so the experimental probability is
P(2 pieces + green salad) = \(\frac{33}{330}\)
= \(\frac{1}{10}\).

The school store sells spiral notebooks in four colors and three different sizes. The table shows the sales by size and color for 400 notebooks.

Question 5.
What is the experimental probability that the next customer will buy a red notebook with 150 pages?
\(\frac{□}{□}\)

Answer: The experimental probability is \(\frac{3}{20}\).

Experiment: The total number of notebooks sold is 400 and in that, red notebooks with 150 pages sold are 60.
So the experimental probability is \(\frac{60}{400}\)
= \(\frac{3}{20}\).

Question 6.
What is the experimental probability that the next customer buys any red notebooks?
\(\frac{□}{□}\)

Answer: The experimental probability is \(\frac{69}{200}\).

Explanation: The total number of notebooks sold is 400 and in that, red notebooks sold are 138.
So the experimental probability that the next customer buys any red notebooks is \(\frac{138}{400}\)
= \(\frac{69}{200}\).

Question 7.
Analyze Relationships
How many combined page counts and color choices are possible? How does this number relate to the number of page size choices and to the number of color choices?

Answer: 12 is the product of the number of page size choices and the number of color choices.

Explanation: As there are 12 entries in the table, there are 12 possible page counts and color combinations. This number relates to the number of page size choices and to the number of color choices by the fact that there are 3-page count choices and 4 colors.
So 3×4= 12.

A middle school English teacher polled random students about how many pages of a book they read per week.

Question 8.
Critique Reasoning
Jennie says the experimental probability that a 7th grade student reads at least 100 pages per week is \(\frac{16}{125}\). What is her error and the correct experimental probability?
\(\frac{□}{□}\)

Answer: The correct experimental probability \(\frac{17}{50}\).

Explanation:
The total number of students is 24+22+30+18+32+53+22+24+25= 250. And the total number of 7th graders that read at least 100 pages is 32+53= 85. Jennie’s error does not include the 7th-grade students who read 150 pages a week. So the experimental probability is
P(7th grade+ al least 100 pages)= \(\frac{85}{250}\)
= \(\frac{17}{50}\).

Question 9.
Analyze Relationships
Based on the data, which group(s) of students should be encouraged to read more? Explain your reasoning.

Answer: The 6th and 8th grades should be encouraged to read more.

Explanation: Based on the data, 6th and 8th grades should be encouraged to read more as 6th and 8th grades read 150 pages per week than 7th grade.

H.O.T. – Page No. 386

Focus on Higher Order Thinking

Question 10.
Make a Conjecture
Would you expect the probability for the simple event “rolling a 6” to be greater than or less than the probability of the compound event “rolling a 6 and getting heads on a coin”? Explain.

Answer: Rolling a 6 to be greater than the probability of the compound event.

Explanation: The simple event would have a greater probability than the probability of the compound event. Because to find a compound event you have to multiply the two probabilities in fraction form. Multiplying two fractions that are less than 1 gives a fraction answer that is smaller than the original two fractions. The probability for the simple event of rolling a 6 is 1/6. The probability of the compound event is 1/6×1/2= 1/12 < 1/6.

Probability Unit Test Grade 7 Answer Key Question 11.
Critique Reasoning
Donald says he uses a standard number cube for simulations that involve 2, 3, or 6 equal outcomes. Explain how Donald can do this.

Answer:
If a simulation has two options A and B, Donald can let the even number be A and the odd number be B. If a stimulation has 3 outcomes A, Band C Donald can let 1 and 2 be A, 3 and 4 be B, and 5 and 6 be C. If a stimulation has 6 outcomes A, B, C, D, E, and F Donald can let 1 be A 2 be B, 3 be C 4 be D 5 be E, and 6 be F

Question 12.
Draw Conclusions
Data collected in a mall recorded the shoe styles worn by 150 male and for 150 female customers. What is the probability that the next customer is male and has an open-toe shoe (such as a sandal)? What is the probability that the next male customer has an open-toe shoe? Are the two probabilities the same? Explain.

Answer: The probability of the next customer is male and has an open-toe shoe is 11/300. And the probability of male customers having open-toe shoes is 11/150.

Explanation:
The total number of customers is 300 and 11 male customers are with open-toe shoes. So the probability of the next customer is male and has an open-toe shoe is 11/300. And the probability of male customers having open-toe shoes is 11/150. The probabilities are not the same, the first one being a compound event and the second one being a simple event.

Question 13.
What If?
Suppose you wanted to perform a simulation to model the shoe style data shown in the table. Could you use two coins? Explain.
______

Answer: No, two coins cannot be used.

Explanation: No, coins cannot be used for this simulation. As there are two options male and female, for the type of customers and two options open and close toe for the type of shoe. It is not given that the customers are equally likely to wear each kind of shoe. So a coin can only be used to simulate male or female.

Question 14.
Represent Real-World Problems
A middle school is made up of grades 6, 7, and 8, and has about the same number of male and female students in each grade. Explain how to use a simulation to find the experimental probability that the first 50 students who arrive at school are male and 7th graders.

Answer: A possible simulation could be done using a coin to simulate a male or female and a standard number of the cube to simulate a grade. Let tails be the male and heads be the female. 1 and 2 be 6th grade, 3 and 4 be 7th grade, and 5 and 6 be the 8th grade. After flipping the coin and rolling the number cube 50 times and recording the results each time and count the number of times you got male and 7th grade out of 50 trails.

Guided Practice – Page No. 390

Question 1.
A baseball player reaches first base 30% of the times he is at bat. Out of 50 times at bat, about how many times will the player reach first base?
______

Answer: So 15 times will the player reach the first base.

Explanation: As the baseball player reaches first base at 30% out of 50 times at bat, so
30% of 50= 0.3×50
= 15.
So 15 times will the player reach the first base.

Question 2.
The experimental probability that it will rain on any given day in Houston, Texas, is about 15%. Out of 365 days, about how many days can residents predict rain?
______

Answer: 55 days can residents predict rain.

Explanation: As the experimental probability that it will rain is 15% out of 365 days, so
15% of 365= 0.15×365
= 54.75
= 55 days.
So 55 days can residents predict rain?

Probability of Compound Events Answers Question 3.
A catalog store has 6% of its orders returned for a refund. The owner predicts that a new candle will have 812 returns out of the 16,824 sold. Do you agree with this prediction? Explain.
______

Answer: The prediction is incorrect.

Explanation: As the catalog store has 6% of its order and 16,824 are sold, so
6% of 16,824 = 0.06×16,824
= 1009 will return.
As the owner predicts that a new candle will have an 812 return which is less than 1009, so the prediction is incorrect.

Question 4.
On a toy assembly line, 3% of the toys are found to be defective. The quality control officer predicts that 872 toys will be found defective out of 24,850 toys made. Do you agree with this prediction? Explain.
______

Answer: The prediction is incorrect.

Explanation: As 3% are found defective out of 24,850 toys, so
3% of 24,850 = 0.03×24850
= 746 will be defective.
As the quality control officer predicts that 872 toys will be found defective which is greater than 746, so the prediction is incorrect.

Question 5.
A light-rail service claims to be on time 98% of the time. Jeanette takes the light-rail 40 times a month, How many times can she predict she will be on time? Is the light-rail’s claim accurate if she is late 6 times?
______

Answer: Jeanette will be on time about 39 times.

Explanation: As light-rail service claims to be on time 98%, and Jeanette takes the light-rail 40 times a month. So
98% of 40= 0.98×40
= 39.
So Jeanette will be on time about 39 times. if she is late 6 times, then the claim is not accurate. Being late 6 times means she was on time 34 times and \(\frac{34}{40}\)= 85% which is not close to 98%.

Question 6.
On average, a college claims to accept 18% of its applicants. If the college has 5,000 applicants, predict how many will be accepted. If 885 applicants are accepted, is the college’s claim accurate?
______

Answer: 900 applicants will be accepted.

Explanation: As the college claims to accept 18% of its applicants of 5,000 applicants, 18% of 5000 is
0.18×5000= 900.
About 900 applicants will be accepted. If 885 applicants are accepted, the claim is accurate because 885 is close to 900.

Essential Question Check-In

Question 7.
How do you make predictions using experimental probability?

Answer: To make a prediction using experimental probability multiply the experimental probability by the number of trials to get the prediction number.

Independent Practice – Page No. 391

The table shows the number of students in a middle school at the beginning of the year and the percentage that can be expected to move out of the area by the end of the year.

Question 8.
How many 7th-grade students are expected to move by the end of the year? If 12 students actually moved, did more or fewer 7th-grade students move than expected? Justify your answer.
______ students

Answer: 8 students from 7th grade are expected to move by the end of the year.

Explanation: As 4% of 7th grades are expected to move by the end of the year, so 4% of 200 is
0.04×200= 8.
If 12 students actually moved, then more than expected would have moved.

Probability and Statistics Worksheets with Answers Pdf Question 9.
Critique Reasoning
The middle school will lose some of its funding if 50 or more students move away in any year. The principal claims he only loses about 30 students a year. Do the values in the table support his claim? Explain.
______

Answer: Yes, the table supports the principal’s claim of 30 students.

Explanation: 2% of 6th graders and 8% of 8th graders are expected to move. So
2% of 250= 0.02×250
= 5.
8% of 150= 0.08×150
= 12
So in total 5+8+12= 25 students are expected to move. And the table supports the principal’s claim of 30 students.

Question 10.
Represent Real-World Problems
An airline knows that, on average, the probability that a passenger will not show up for a flight is 6%. If an airplane is fully booked and holds 300 passengers, how many seats are expected to be empty? If the airline overbooked the flight by 10 passengers, how many passengers are expected to show up for the flight? Justify your answer.
______

Answer: The number of passengers expected to show up is 310-19= 291 passengers.

Explanation: As 6% of the 300 seats are expected to be empty, so
6% of 300 = 0.06×300
= 18.
18 seats are expected to be empty. If the airline overbooked the flight by 10 passengers then 300+10= 310 passengers were booked, then 310×0.06= 18.6
= 19
So the number of passengers expected to show up is 310-19= 291 passengers.

Question 11.
Draw Conclusions
In a doctor’s office, an average of 94% of the clients pay on the day of the appointment. If the office has 600 clients per month, how many are expected not to pay on the day of the appointment? If 40 clients do not pay on the day of their appointment in a month, did more or fewer than the average not pay?
______

Answer:

Explanation: 94% of the clients pay on the day of the appointment for 600 clients, so
94% of 600= 0.94 × 600
= 564
As 564 clients are expected to pay 600- 564= 36 clients are expected not to pay on the day of the appointment. So if 40 clients do not pay, then this a little more than the average.

Page No. 392

Question 12.
Counterexamples
The soccer coach claimed that, on average, only 80% of the team came to practice each day. The table shows the number of students that came to practice for 8 days. If the team has 20 members, how many team members should come to practice to uphold the coach’s claim? Was the coach’s claim accurate? Explain your reasoning.

Answer: As all the values in the table are greater than 16 except for one value, the claim is not accurate.

Explanation: As the soccer coach claimed that only 80% of the team come to practice each day, and the team has 20 members. So
80% of 20= 0.8×20
= 16.
So 16 players in the team should come to practice to uphold the coach’s claim. As all the values in the table are greater than 16 except for one value, the claim is not accurate. More than 80% come on average to practice each day and more than 16 members on average come to practice.

Question 13.
What’s the Error?
Ronnie misses the school bus 1 out of every 30 school days. He sets up the proportion \(\frac{1}{30}\) = \(\frac{180}{x}\) to predict how many days he will miss the bus in the 180-day school year. What is Ronnie’s error?

Answer: The proportion he sets up is \(\frac{1}{30}\) = \(\frac{180}{X}\) is Ronnie’s error.

Explanation: As Ronnie misses the school bus 1 out of every 30 school days, and the proportion he sets up is \(\frac{1}{30}\) = \(\frac{180}{X}\) which is incorrect. As it should be \(\frac{1}{30}\) = \(\frac{X}{180}\) to respect the form of \(\frac{number of days missed}{total number of days}\)

H.O.T.

Focus on Higher Order Thinking

Question 14.
Persevere in Problem-Solving
A gas pump machine rejects 12% of credit card transactions. If this is twice the normal rejection rate for a normal gas pump, how many out of 500 credit cards transactions would a normal gas pump machine reject?
______

Answer: The normal gas machine rejects 30 transactions.

Explanation: If 12% is twice the normal rate, then the normal rate is 6%, so 6% of 500 is
0.06×500= 30.
So the normal gas machine rejects 30 transactions.

Question 15.
Make Predictions
An airline’s weekly flight data showed a 98% probability of being on time. If this airline has 15,000 flights in a year, how many flights would you predict to arrive on time? Explain whether you can use the data to predict whether a specific flight with this airline will be on time.
______

Answer: 14,700 flights we can predict to arrive on time.

Explanation: The airline’s weekly flight data is a 98% probability of being on time, and there are 15,000 flights in a year. So 98% of 15,000 is 0.98×15000 = 14,700.
So 14,700 flights are expected to be on time and 15,000-14,700= 300 flights will not be on time. And we can use the data to predict whether a specific flight with this airline will be on time by determining the number of flights that have not been on time. If about 300 flights have not been on time then we can predict that the specific flight will be on time. If less than 300 flights have not been on time, then you can’t predict with absolute certainty if the specific flight will be on time.

Question 16.
Draw Conclusions
The average response rate for a marketing letter is 4%, meaning that 4% of the people who receive the letter respond to it. A company writes a new type of marketing letter, sends out 2,400 of them, and gets 65 responses. Explain whether the new type of letter would be considered to be a success.

Answer: As we have received fewer responses than expected and thus a new type of letter would be considered to be not successful.

Explanation: The company sends 2400 letters. And the average response rate for marketing is 4%, so we can expect only a 4% response to the 2400 letters. So
4% of 2400= \(\frac{4}{100}\) × 2400
= 4×24
= 96.
Since 96 is greater than 65, we have received fewer responses than expected and thus a new type of letter would be considered to be not successful. We should receive more responses than expected if the letter was successful.

12.1 Probability – Page No. 393

Question 1.
Josue tosses a coin and spins the spinner at the right. What are all the possible outcomes?

Answer: The possible outcomes are heads and 1, heads and 2, tails and 1, tails and 2.

Explanation: The spinner can land on 1 or 2 so the outcomes for the spinner 1 and 2. So the coin can land on heads or tails so the outcomes for the coin are heads and tails. The outcomes for tossing a coin and spinning the spinner are heads and 1, heads and 2, tails and 1, tails and 2.

12.2 Experimental Probability of Simple Events

Question 2.
While bowling with friends, Brandy rolls a strike in 6 out of 10 frames. What is the experimental probability that Brandy will roll a strike in the first frame of the next game?
\(\frac{□}{□}\)

Answer: The experimental probability is \(\frac{3}{5}\).

Explanation: As there are 10 frames, so possible outcomes are 10. And Brandy rolls a strike in 6, so the experimental probability that Brandy will roll a strike in the first frame of the next game is \(\frac{6}{10}\)= \(\frac{3}{5}\).

Section 12.2 Experimental Probability Worksheet Question 3.
Ben is greeting customers at a music store. Of the first 20 people he sees enter the store, 13 are wearing jackets and 7 are not. What is the experimental probability that the next person to enter the store will be wearing a jacket?
\(\frac{□}{□}\)

Answer: The experimental probability \(\frac{13}{20}\).

Explanation: The total number of people is 20, and Ben sees 13 people were wearing jackets. So the experimental probability that the next person to enter the store will be wearing a jacket is \(\frac{13}{20}\).

12.3 Experimental Probability of Compound Events

Question 4.
Auden rolled two number cubes and recorded the results.

What is the experimental probability that the sum of the next two numbers rolled is greater than 5?
\(\frac{□}{□}\)

Answer: The experimental probability is latex]\frac{3}{7}[/latex].

Explanation:
The sum of two numbers for every roll is
Roll #1 is 2+1= 3.
Roll #2 is 4+5= 9.
Roll #3 is 3+2= 5.
Roll #4 is 2+2= 4.
Roll #5 is 1+3= 4
Roll #6 is 6+2= 8.
Roll #7 is 5+3= 8.
As there are 7 rolls, so the possible outcomes are 7, so the experimental probability that the sum of the next two numbers rolled is greater than 5 is \(\frac{3}{7}\).

12.4 Making Predictions with Experimental Probability

Question 5.
A player on a school baseball team reaches first base \(\frac{3}{10}\) of the time he is at bat. Out of 80 times at bat, about how many times would you predict he will reach first base?
______

Answer: We can predict that he will reach first base 24 times out 80 times at the bat.

Explanation:
We need to write a proportion:
latex]\frac{3}{10}[/latex]= latex]\frac{X}{80}[/latex].
10×X= 80×3
10×X= 240
X= 24.
So we can predict that he will reach first base 24 times out 80 times at the bat.

Essential Question

Question 6.
How is experimental probability used to make predictions?

Answer:

Selected Response – Page No. 394

Question 1.
A frozen yogurt shop offers scoops in cake cones, waffle cones, or cups. You can get vanilla, chocolate, strawberry, pistachio, or coffee flavored frozen yogurt. If you order a single scoop, how many outcomes are in the sample space?
Options:
a. 3
b. 5
c. 8
d. 15

Answer: The possible outcomes are 3×5= 15.

Explanation: There is three option for the scoops cake cones, waffle cones, or cups. And the five flavors are vanilla, chocolate, strawberry, pistachio, or coffee. So the possible outcomes are 3×5= 15.

Question 2.
A bag contains 7 purple beads, 4 blue beads, and 4 pink beads. What is the probability of not drawing a pink bead?
Options:
a. \(\frac{4}{15}\)
b. \(\frac{7}{15}\)
c. \(\frac{8}{15}\)
d. \(\frac{11}{15}\)

Answer: The probability of not drawing a pink bead is \(\frac{11}{15}\).

Explanation: The total number of beads in the bag are 7+4+4= 15 beads. And the pink beads are 4 beads.
The probability of not drawing a pink bead is
P(pink)+P(not pink)= 1
\(\frac{4}{15}\)+ P(not pink)= 1
P(not pink)= 1- \(\frac{4}{15}\)
= \(\frac{15-4}{15}\)
= \(\frac{11}{15}\).

Question 3.
During the month of June, Ava kept track of the number of days she saw birds in her garden. She saw birds on 18 days of the month. What is the experimental probability that she will see birds in her garden on July 1?
Options:
a. \(\frac{1}{18}\)
b. \(\frac{2}{5}\)
c. \(\frac{1}{2}\)
d. \(\frac{3}{5}\)

Answer: The experimental probability that she will see birds in her garden on July 1 is \(\frac{3}{5}\).

Explanation:
As there are 30 days in the month of June, so if Ava saw birds of those days, the experimental probability that she will see birds in her garden on July 1 is \(\frac{18}{30}\)= \(\frac{3}{5}\).

Lesson 3 Homework Practice Probability of Compound Events Answer Key Question 4.
A rectangle has a width of 4 inches and a length of 6 inches. A similar rectangle has a width of 12 inches. What is the length of a similar rectangle?
Options:
a. 8 inches
b. 12 inches
c. 14 inches
d. 18 inches

Answer: The length is 18 inches.

Explanation: The length of the rectangle is 6 inches and the width is 4 inches similarly width of the other rectangle is 12 inches so the length is
\(\frac{Length}{Width}\)= \(\frac{6}{4}\)= \(\frac{X}{12}\).
4×X= 12×6
4X= 72
X= 18 inches.

Question 5.
The experimental probability of hearing thunder on any given day in Ohio is 30%. Out of 600 days, on about how many days can Ohioans expect to hear thunder?
Options:
a. 90 days
b. 180 days
c. 210 days
d. 420 days

Answer: The number of days is 180 days.

Explanation: We need to find 30%. Out of 600 days
= 0.3×600
= 180 days.
The number of days is 180 days.

Question 6.
Isidro tossed two coins several times and then recorded the results in the table below.

What is the experimental probability that both coins will land on the same side on Isidro’s next toss?
Options:
a. \(\frac{1}{5}\)
b. \(\frac{2}{5}\)
c. \(\frac{3}{5}\)
d. \(\frac{4}{5}\)

Answer: The experimental probability that both coins will land on the same side on Isidro’s next toss is \(\frac{2}{5}\).

Explanation: As there are 5 tosses and possible outcomes are 5. As the coin was landed twice on the same side, so the experimental probability is \(\frac{2}{5}\).

Mini-Task

Question 7.
Magdalena had a spinner that was evenly divided into sections of red, blue, and green. She spun the spinner and tossed a coin several times. The table below shows the results.

a. What are all the possible outcomes?

Answer: The possible outcomes are RH,RT,BH,BT,GH,GT.

Explanation:
The spinner can land on red, blue, and green and the coin can land on heads or tails so the possible outcomes are red and heads, red and tails, blue and heads, blue and tails, green and heads, and green and tails.

Question 7.
b. What experimental probability did Magdalena find for spinning blue? Give your answer as a fraction in simplest form, as a decimal, and as a percent.

Answer: The experimental probability is 40%.

Explanation: The total trails are 5 and Magdalena spun blue twice,
so the experimental probability is 2/5 = 0.4
= 40%

Question 7.
c. Out of 90 trials, how many times should Magdalena predict she will spin green while tossing tails?
______ times

Answer: We can predict that she will spin green 36 times.

Explanation: The total trails are 5 and Magdalena spun green twice,
so the experimental probability is 2/5.
And out of 90 trails, we can predict that she will spin green
2/5×90
= 18×2
= 36 times.

Conclusion:

Prepare in Go Math Answer Key and improve your math skills. You can also check your skills after preparation by solving the questions which are provided at the end of the chapter.

Guided Practice – Page No. 268

Find the circumference of each circle.

Question 1.

________ in

Answer: 56.57 in

Explanation:
Circumference of the circle = 2πr = 2 x 22/7 x 9 = 56.57 in

Question 2.

________ cm

Answer: 44 cm

Explanation:
Circumference of the circle = 2πr = 2 x 22/7 x 7 = 44 cm

Find the circumference of each circle. Use 3.14 or \(\frac{22}{7}\) for π. Round to the nearest hundredth, if necessary.

Question 3.
______ m

Question 4.

______ yd

Answer: 30.15 yd

Explanation:
Circumference of the circle = 2πr = 2 x 3.14 x 4.8 = 30.144 yd

Question 5.

______ in

Answer: 7.5 in

Explanation:
Circumference of the circle = 2πr = 2 x 3.14 x 7.5 = 47.1 in

9.1 Circles and Circumference Answer Key Question 6.
A round swimming pool has a circumference of 66 feet. Carlos wants to buy a rope to put across the diameter of the pool. The rope costs $0.45 per foot, and Carlos needs 4 feet more than the diameter of the pool. How much will Carlos pay for the rope?
$ ______

Answer: $6.525

Explanation:
Circumference of the swimming pool = 66 feet
πd = 66
22/7 x d = 66
d = 66 x 7/ 22 = 10.5
The diameter of the pool = 10.5 feet
Carlos needs 4 feet more than the diameter of the pool.
Total rope needed = 10.5 + 4 = 14.5 feet
Cost of rope per foot = $0.45
Total cost of the rope = 14.5 x $0.45 = $6.525
Therefore the total cost of the rope = $6.525

Find each missing measurement to the nearest hundredth. Use 3.14 for π.

Question 7.
r =
d =
C = π yd
r = ________ yd
d = ________ yd

Answer:
r = 0.5 yd
d = 1 yd

Explanation:
Circumference = π yd
2πr = π yd
r = 1/2 yd = 0.5 yd
d = 2r = 2 [1/2] = 1 yd

Question 8.
r ≈
d ≈
C = 78.8 ft
r ≈ ________ ft
d ≈ ________ ft

Answer:
r = 495.31 ft
d = 990.62 ft

Explanation:
Circumference = 78.8 ft
2πr = 78.8 ft
r = 2 x 22/7 x 78.8 = 495.31 ft
d = 2 x 495.31 = 990.62 ft

Question 9.
r ≈
d ≈ 3.4 in
C =
r ≈ ________ in
C = ________ in

Answer:
r = 1.7 in
c = 10.68 in

Explanation:
Diameter = 3.4 in
Circumference = πd = 22/7 x 3.4 in = 10.68 in
r = d/2 = 1.7 in

Essential Question Check-In

Question 10.
Norah knows that the diameter of a circle is 13 meters. How would you tell her to find the circumference?
Type below:
____________

Answer: Circumference = 16.82 meters

Explanation:
Given,
Diameter = 13 meters
Circumference = πd = 22/7 x 13 = 16.82 meters

Independent Practice – Page No. 269

For 11–13, find the circumference of each circle. Use 3.14 or \(\frac{22}{7}\) for π. Round to the nearest hundredth, if necessary.

Question 11.

_______ ft

Answer:
Circumference = 18.526 ft = 19 ft (approx)

Explanation:
Given:
Diameter = 5.9 ft
Circumference = πd = 3.14 x 5.9 = 18.526 ft = 19 ft (approx)

9.1 Circles and Circumference Answer Key Question 12.

_______ cm

Answer:
Circumference =176 cm

Explanation:
Given:
Radius = 56 cm
Circumference = πd = 22/7 x 56 = 176 cm

Question 13.

_______ in

Answer:
Circumference = 110 in

Explanation:
Given:
Diameter = 35 in
Circumference = πd = 22/7 x 35 = 110 in

Question 14.
In Exercises 11–13, for which problems did you use \(\frac{22}{7}\) for π? Explain your choice.
Type below:
_____________

Answer:
The 11th question is 3.14 and the 12 and 13 questions as π

Explanation:
We can take 3.14 as π for 11 th question because the diameter is given in decimal points.
And in questions 12 and 13 we need to take π because the radius and diameter are given in whole number form.

Question 15.
A circular fountain has a radius of 9.4 feet. Find its diameter and circumference to the nearest tenth.
d = _________ ft
C = _________ ft

Answer:
d = 19 ft
C = 59 ft

Explanation:
Given:
Radius = 9.4 ft
Diameter = 2r = 2 x 9.4  = 18.8 ft = 19 ft (approx)
Circumference = πd = 22/7 x 18.8 = 59.08 = 59 ft (approx)

Question 16.
Find the radius and circumference of a CD with a diameter of 4.75 inches.
r = _________ in
C = _________ in

Answer:
r = 2.4 in
C = 15 in

Explanation:
Given:
Diameter = 4.75 in
Radius = r/2 = 4.75/2 = 2.37 in = 2.4 in (approx)
Circumference = πd = 22/7 x 4.75 = 14.92 in =15 in (approx)

Question 17.
A dartboard has a diameter of 18 inches. What are its radius and circumference?
r = _________ in
C = _________ in

Answer:
r = 9 in
C = 56.6 in

Explanation:
Given:
Diameter = 18 in
Radius = r/2 = 18/2 = 9 in
Circumference = πd = 22/7 x 18 = 56.57 in = 56.6 in (approx)

Question 18.
Multistep
Randy’s circular garden has a radius of 1.5 feet. He wants to enclose the garden with edging that costs $0.75 per foot. About how much will the edging cost? Explain.
$ _______

Answer:

Explanation:
Given:
The radius of the garden= 1.5 ft
Circumference of the garden = 2πr = 2 x 22/7 x 1.5 = 9.42 ft
The cost of enclosing the garden per foot = $0.75
Total cost of edging = 9.42 x $0.75 = $7.06 = $7 (approx)

Question 19.
Represent Real-World Problems
The Ferris wheel shown makes 12 revolutions per ride. How far would someone travel during one ride?

_______ ft

Answer: Total distance travelled in one ride is 4,752 ft

Explanation:
Given:
The diameter of the Ferris wheel= 63 ft
Circumference of the Ferris wheel = 2πr = 2 x 22/7 x 63 = 396 ft
Total number of revolutions = 12
Total distance travelled = 12 x 396 = 4,752 ft

Question 20.
The diameter of a bicycle wheel is 2 feet. About how many revolutions does the wheel make to travel 2 kilometers? Explain. Hint: 1 km ≈ 3,280 ft
_______ revolutions

Answer:
1044 revolutions

Explanation:
Given:
The diametre of the bicycle wheel = 2 feet
Total distance travelled = 2 kilometres
We know that,
1 km ≈ 3,280 ft
2 km = 2 x 3,280 = 6,560 ft
Circumference of the bicycle = Distance travelled in one revolution = πd = 22/7 x 2 = 6.28 ft = 6.3 ft
Total number of revolutions = Total distance traveled / distance traveled in one revolution
= 6560 / 6.28 = 1044  revolutions

9.1 Independent Practice Question 21.
Multistep
A map of a public park shows a circular pond. There is a bridge along the diameter of the pond that is 0.25 mi long. You walk across the bridge, while your friend walks halfway around the pond to meet you at the other side of the bridge. How much farther does your friend walk?
_______ mi

Answer:

Explanation:
Given,
The diameter of the pond = 0.25 mi
The length of the bridge = The diameter of the pond = 0.25 mi
Then the distance walked by the man = 0.25 mi
Distance travelled by the friend = Halfway around the pond to meet you at the other side of the bridge = πd/2
= 22/7 x 0.25/2  = 0.39 = 0.4 mi
The friend travelled more distance compared to the man
The more distance travelled by the friend = 0.39 – 0.25 = 0.14 mi

Page No. 270

Question 22.
Architecture
The Capitol Rotunda connects the House and the Senate sides of the U.S. Capitol. Complete the table. Round your answers to the nearest foot.

Type below:
_____________

Answer:
Radius = 48 ft
Diameter = 96 ft

Explanation:
Given
Height = 180 ft
Circumference = 301.5 ft
πd = 301.5
22/7 x d = 301.5
d = 95.93 = 96 ft
r = d/2 = 96/2 = 48 ft

H.O.T.

Focus on Higher Order Thinking

Question 23.
Multistep
A museum groundskeeper is creating a semicircular statuary garden with a diameter of 30 feet. There will be a fence around the garden. The fencing costs $9.25 per linear foot. About how much will the fencing cost altogether?
$ _______

Answer:
The total cost of fencing = $712

Explanation:
Given,
The diameter = 30 ft
Circumference of the garden in the shape of circle = 2πr
Circumference of the semicircle = πr = πd/2 =  22/7 x 30/2 = 47.14ft
Cost of fencing for each foot = $9.25
The total cost of fencing the semicircular garden = 47.14 x $9.25 + 30 x  $9.25  = $712 (approx)

Question 24.
Critical Thinking
Sam is placing rope lights around the edge of a circular patio with a diameter of 18 feet. The lights come in lengths of 54 inches. How many strands of lights does he need to surround the patio edge?
_______ strands

Answer: 12 and a half strands of light = 13 strands (approx)

Explanation:
Given,
The diameter of the circular patio = 18 ft = 216 inch
Circumference of the circular patio = πd = 22/7 x 216 = 678.85 inch
The lights will come in a length (in one strand)= 54 inches
Total number of strands of light required for the circular patio
= Circumference of the circular patio/ The lights will come in a length (in one strand) = 678.85/54 = 12.57 = 12 and a half strands of light

Question 25.
Represent Real-World Problems
A circular path 2 feet wide has an inner diameter of 150 feet. How much farther is it around the outer edge of the path than around the inner edge?
_______ feet

Answer: about 12.6 ft

Explanation:
Given,
Width of the circular path = 2 ft
The inner diameter of the circular path = 150 ft
The outer diameter of the circular path = 150 + 2(2) = 154 ft
Inner circumference = πd = 150 π
Outer circumference =  πd = 154π
Distance between the outer and inner edge = 154 π – 150 π = 4 π = 12.6 ft

Circumference Area and Volume Class 7 Question 26.
Critique Reasoning
The gear on a bicycle has the shape of a circle. One gear has a diameter of 4 inches, and a smaller one has a diameter of 2 inches. Justin says that the circumference of the larger gear is 2 inches more than the circumference of the smaller gear. Do you agree? Explain your answer.
_______

Answer:
Justin’s statement is incorrect.

Explanation:
The circumference of the larger gear = πd = 4π
The circumference of the smaller gear = πd = 2π
Since, 2 x 2π = 4π, the circumference of the larger gear is two times the circumference of the smaller gear.
Since = 4π – 2π = 2π = 6.28
Therefore, The larger circumference is not 2 inches more than the smaller circumference

Question 27.
Persevere in Problem-Solving
Consider two circular swimming pools. Pool A has a radius of 12 feet, and Pool B has a diameter of 7.5 meters. Which pool has a greater circumference? How much greater? Justify your answers.
_______

Answer:
Pool B about 0.9 meters

Explanation:
Given,
Pool A has a diameter = 24 ft
Pool B has a diameter = 7.5 m
We know that,
1 ft = 0.3 metres
24 ft = 7.2 metres
Pool B has a greater diameter so it has a greater circumference.
Circumference of the pool A = 7.2π
Circumference of the pool B = 7.5π
Difference between the circumferences = 7.5π – 7.2π = 0.9 meters.

Guided Practice – Page No. 274

Find the area of each circle. Round to the nearest tenth if necessary. Use 3.14 for π.

Question 1.

_______ m²

Answer: 153.9 m²

Explanation:
Given:
Diameter = 14 m
Radius = 14/2 = 7 m
Area of the circle = πr²
= 3.14 x 7 x 7 = 153.86 = 153.9 m²

Question 2.

_______ mm²

Answer: 452.2 mm²

Explanation:
Given:
Radius =12mm
Area of the circle = πr²
= 3.14 x 12 x 12 = 3.14(144) = 452.2mm²

Question 3.

_______ yd²

Answer: 314 yd²

Explanation:
Given:
Diameter = 20yd
Radius = 20/2 = 10yd
Area of the circle = πr²
= 3.14 x 10 x 10 = 3.14(100) = 314yd²

Solve. Use 3.14 for π.

Question 4.
A clock face has a radius of 8 inches. What is the area of the clock face? Round your answer to the nearest hundredth.
_______ in²

Answer: 200.96 in²

Explanation:
Given:
Radius = 8inches
Area of the clock face = πr²
= 3.14 x 8 x 8= 3.14(64) = 200.96 in²

Chapter 9 Circumference, Area and Volume Answer Key Question 5.
A DVD has a diameter of 12 centimeters. What is the area of the DVD? Round your answer to the nearest hundredth.
_______ cm²

Answer: 113.04 cm²

Explanation:
Given:
Diameter = 12 centimeters
Radius = 12/2 = 6 centimeters
Area of the DVD= πr²
= 3.14 x 6 x 6 = 3.14(36) = 113.04 cm²

Question 6.
A company makes steel lids that have a diameter of 13 inches. What is the area of each lid? Round your answer to the nearest hundredth.
_______ in²

Answer: 132.67 in²

Explanation:
Given:
Diameter = 13 inches
Radius = 13/2 = 6.5 inches
Area of each lid= πr²
= 3.14 x 6.5 x 6.5 = 3.14(42.25) = 132.67 in²

Find the area of each circle. Give your answers in terms of π.

Question 7.
C = 4π
A =
Type below:
______________

Answer: 4π

Explanation:
Given:
Circumcenter = 4π
2πr = 4π
Radius = 4/2 = 2 units
Area of the circle = πr²
= π x 2 x 2 = π(4) = 4π square units

Question 8.
C = 12π
A =
Type below:
______________

Answer: 36π

Explanation:
Given:
Circumcenter = 12π
2πr = 12π
Radius =6 units
Area of the circle = πr²
= π x 6 x 6 = π(36) = 36π square units

Question 9.
C = \(\frac{π}{2}\)
A =
Type below:
______________

Answer: π/16

Explanation:
Given:
Circumcenter = \(\frac{π}{2}\)
2πr = \(\frac{π}{2}\)
Radius = 1/4 units
Area of the circle = πr²
= π x 1/4 x 1/4 = π(1/16) = π/16 square units

Lesson 9.2 Area of Circles Practice and Problem-Solving a/b Answers Question 10.
A circular pen has an area of 64π square yards. What is the circumference of the pen? Give your answer in terms of π
Type below:
______________

Answer: 16π

Explanation:
Given:
Area of the circular pen = 64π square yards
πr² = 64π
r = 8 yards
Circumference of the circle = 2πr = 2 x 8 x π = 16π yards

Essential Question Check-In

Question 11.
What is the formula for the area A of a circle in terms of the radius r?
Type below:
______________

Answer: πr²

Explanation:
Area of a circle = πr²

Independent Practice – Page No. 275

Question 12.
The most popular pizza at Pavone’s Pizza is the 10-inch personal pizza with one topping. What is the area of a pizza with a diameter of 10 inches? Round your answer to the nearest hundredth.
_______ in²

Answer: 78.5 in²

Explanation:
Given:
Diameter = 10 inches
Radius = 10/2 = 5 inches
Area of a pizza = πr²
= 3.14 x 5 x 5 = 3.14(25) = 78.5 in²

Question 13.
A hubcap has a radius of 16 centimeters. What is the area of the hubcap? Round your answer to the nearest hundredth.

_______ cm²

Answer: 803.84 cm²

Explanation:
Given:
Radius = 16 cm
Area of the circle = πr²
= 3.14 x 16 x 16 = 3.14(256) = 803.84 cm²

Question 14.
A stained glass window is shaped like a semicircle. The bottom edge of the window is 36 inches long. What is the area of the stained glass window? Round your answer to the nearest hundredth.
_______ in²

Answer: 508.68 in²

Explanation:
Area of the semicircle = 1/2 πr² = 1/2(3.14)(18)(18) = 1/2 (3.14)(324) = 1.57(324) = 508.68 in ²

Question 15.
Analyze Relationships
The point (3,0) lies on a circle with the centre at the origin. What is the area of the circle to the nearest hundredth?
_______ units²

Answer: 28.26 units²

Explanation:
Radius = 3
Area of the circle = πr² = π(3)² = 3.14(9) = 28.26 units²

Question 16.
Multistep
A radio station broadcasts a signal over an area with a radius of 50 miles. The station can relay the signal and broadcast over an area with a radius of 75 miles. How much greater is the area of the broadcast region when the signal is relayed? Round your answer to the nearest square mile.
_______ mi²

Answer: 9813 mi²

Explanation:
Given:
The radius of a radio station broadcasting the signal (r) = 50 miles
The greatest radius to which the broadcast can be relayed (R) = 75 miles
The greatest area of the broadcast region when the signal is relayed = πR²-πr² = π(75) (75) – π (50) (50)
= 5625π – 2500π
= 3125π
= 3125(3.14) = 9813 mi²(approx)

Chapter 9 Practice Test Surface Area Question 17.
Multistep
The sides of a square field are 12 meters. A sprinkler in the center of the field sprays a circular area with a diameter that corresponds to a side of the field. How much of the field is not reached by the sprinkler? Round your answer to the nearest hundredth.
_______ m²

Answer:30.96 m²

Explanation:
Given:
The side of the square = 12 meters
The diameter circular area of the field in the centre = The side of the square = 12 meters
The radius of the field = 12/2 = 6 meters
Area of the field which is not reached by the sprinkler = Area of the square – Area of the circular area
= (side)²-πr² = (12)(12) – π (6) (6)
= 144 – 36 (3.14)
= 144 – 113.04
= 30.96 m²

Question 18.
Justify Reasoning
A small silver dollar pancake served at a restaurant has a circumference of 2π inches. A regular pancake has a circumference of 4π inches. Is the area of the regular pancake twice the area of the silver dollar pancake? Explain.
_______

Answer: No, the area of the regular pancake is 4 times the area of the silver dollar pancake

Explanation:
Silver Dollar pancake:
Circumference of the silver Dollar pancake = 2π inches
2πr = 2π
r = 1 inch
Area of the silver dollar pancake = πr² = π (1) (1) = π in²

Regular pancake:
Circumference of the regular pancake = 4π inches
2πr = 4π
r = 2 inch
Area of the silver dollar pancake = πr² = π (2) (2) = 4π in²

Therefore, the area of the regular pancake is 4 times the area of the silver dollar pancake

Question 19.
Analyze Relationships
A bakery offers a small circular cake with a diameter of 8 inches. It also offers a large circular cake with a diameter of 24 inches. Does the top of the large cake have three times the area of that of the small cake? If not, how much greater is its area? Explain.
_______

Answer: No, the area of the large cake is 9 times the area of the small cake

Explanation:
Small Cake:
The diameter of the small cake= 8 inches
The radius of the small cake = 8/2 = 4 inches
Area of the small cake  = πr² = π (4) (4) = 16 π in²

Large Cake:
The diameter of the large cake= 24 inches
The radius of the large cake = 24/2 = 12 inches
Area of the large cake  = πr² = π (12) (12) = 144 π in²

Since 144 π/ 16 π = 9
Therefore the

area of the large cake is 9 times the area of the small cake.

Page No. 276

Question 20.
Communicate Mathematical Ideas
You can use the formula A = \(\frac{C^{2}}{4π}\) to find the area of a circle given the circumference. Describe another way to find the area of a circle when given the circumference.
Type below:
____________

Answer: Area = C²/4π

Explanation:
Circumference of the circle = 2πr
C = 2πr
Divide both sides by 2π
then, r = C/2π
Area of the circle = πr²
Substitute C/2π for r:
Area = π(c/2π)² = C²/4π

Question 21.
Draw Conclusions
Mark wants to order a pizza. Which is the better deal? Explain.

_____

Answer: The pizza of 18 inches is a better deal

Explanation:
Given:
The diameter of the pizza = 12 inches
The radius of the pizza = 12/2= 6 inches
Area of the circle = πr²
= (3.14)(6)(6) = 113 (approx) in²
The total cost of the pizza = $10
Cost of the pizza per inch = $10/113 = $0.09 per square inch

The diameter of the pizza = 18 inches
The radius of the pizza = 18/2= 9 inches
Area of the circle = πr²
= (3.14)(9)(9) = 254 (approx) in²
The total cost of the pizza = $20
Cost of the pizza per inch = $20/254 = $0.08 per inch

Question 22.
Multistep
A bear was seen near a campground. Searchers were dispatched to the region to find the bear.
a. Assume the bear can walk in any direction at a rate of 2 miles per hour. Suppose the bear was last seen 4 hours ago. How large an area must the searchers cover? Use 3.14 for π. Round your answer to the nearest square mile.
_____ mi²

Answer: 201mi²

Explanation:
The bear can walk a distance = 2 x 4 = 8 miles
Since it is walking 2 miles per hour for 4 hours
The radius of the bear = 8 miles
Area of the circle = πr²
= (3.14)(8)(8) = 201 (approx) mi²

Question 22.
b. What If? How much additional area would the searchers have to cover if the bear were last seen 5 hours ago?
_____ mi²

Answer: 113mi²

Explanation:
If the bear for 5 hours then,
The bear can walk a distance = 2 x 5 = 10 miles
Since it is walking 2 miles per hour for 5 hours
The radius of the bear = 10 miles
Area of the circle = πr²
= (3.14)(10)(10) = 314 (approx) mi²

The additional area covered by the searches = 314 – 201 = 113 mi²

H.O.T.

Focus on Higher Order Thinking

Question 23.
Analyze Relationships
Two circles have the same radius. Is the combined area of the two circles the same as the area of a circle with twice the radius? Explain.
_____

Answer: No

Explanation:
If the radius of two circles is the same.
then, Let the radii of the circles be 1.
The area of each circle =  π square units
The combined area of 2 circles =π+π = 2π square units

If the radius is doubled.
then, Let the radii of the circles be 2
The area of each circle =  4π square units
The combined area of 2 circles =  4π+4π = 8π square units

Therefore the areas of both cases are not the same.

Question 24.
Look for a Pattern
How does the area of a circle change if the radius is multiplied by a factor of n, where n is a whole number?
Type below:
____________

Answer: The new area is then n² times the area of the original circle.

Explanation:
If the radius is multiplied by a factor “n”
then, the new radius = rn
The area of the circle (with radius rn) = π(rn)² = n² (πr²).
Therefore the new area is n² times the area of the original circle.

Question 25.
Represent Real World Problems
The bull’s-eye on a target has a diameter of 3 inches. The whole target has a diameter of 15 inches. What part of the whole target is the bull’s-eye? Explain.
Type below:
____________

Answer: 1/25 of the target

Explanation:
Bull’s eye:
Diameter of Bull’s eye = 3 inches
Radius of Bull’s eye = 3/2 = 1.5 inches
Area of the Bull’s eye = π(r)² = π(1.5)² = 2.25π
Target:
Diameter of the target = 15 inches
Radius of the target = 15/2 = 7.5 inches
Area of the target = π(r)² = π(7.5)² = 56.25π

The part of Bull’s eye in the whole target = 2.25π/ 56.25π = 1/25

Therefore the 1/25th part of the whole target is the Bull’s eye.

Guided Practice – Page No. 280

Question 1.
A tile installer plots an irregular shape on grid paper. Each square on the grid represents 1 square centimeter. What is the area of the irregular shape?

_____ cm²

Answer: Area of the irregular shape = 34 cm²

Explanation:
STEP1 First divide the irregular shapes into polygons.
STEP2 The irregular shape can be divided into a triangle, rectangle, parallelogram
STEP3 Areas of the polygons
Area of triangle = 1/2 (base x height) = 1/2 (4 x 2) = 4 cm²
Area of the rectangle = length x breadth = 5 x 3 = 15 cm²
Area of the parallelogram = base x height = 5 x 3 = 15 cm²
Area of the irregular shape = (15+15+5) cm²= 34cm²

Question 2.
Show two different ways to divide the composite figure. Find the area both ways. Show your work below.

_____ cm²

Answer: Area of the figure in both ways = 288 cm²

Explanation:
The first way to divide up the composite shape is to divide it into an 8 by 9 rectangle and a 12 by 18 rectangle.
The area of the first rectangle = Length x breadth = 9 x 8 = 72 cm²
The area of the second rectangle =  Length x breadth = 18 x 12 = 216 cm²
The total area of the figure = 72 + 216 = 288 cm²

Question 3.
Sal is tiling his entryway. The floor plan is drawn on a unit grid. Each unit length represents 1 foot. Tile costs $2.25 per square foot. How much will Sal pay to tile his entryway?

$ _____

Answer: Sal will pay $97.875

Explanation:
Separate this figure into trapezium and parallelogram.
Area of the trapezium = 1/2 (a+b)h = 1/2 (7+4) 5 = 1/2 (11) 5 = 27.5 ft²
Area of the parallelogram = base x height = 4 x 4 = 16 ft²

The total area of the figure = 27.5 + 16 = 43.5ft²
Cost of each square foot = $2.25
Amount paid by Sal = 43.5 x 2.25 = $97.875

Essential Question Check-In

Question 4.
What is the first step in finding the area of a composite figure?
Type below:
______________

Answer:
The first step in finding the area of a composite figure is to divide it up into smaller basic shapes.

Explanation:
The first step in finding the area of a composite figure is to divide it up into smaller basic shapes such as triangles, squares, rectangles, parallelograms, circles and trapezium.
Then calculate the area of each figure and add them to find the area of the figure.

Independent Practice – Page No. 281

Question 5.
A banner is made of a square and a semicircle. The square has side lengths of 26 inches. One side of the square is also the diameter of the semicircle. What is the total area of the banner? Use 3.14 for π.

_____ in²

Answer: 941.33 in²

Explanation:
Area of the square = side x side = 26 x 26 = 676 in²
Area of the semicircle =1/2 πr²= 1/2 (3.14) (13) (13) = 1/2 (3.14) (169) = 265.33 in²
Area of the figure = 676 + 265.33 = 941.33 in²

Question 6.
Multistep
Erin wants to carpet the floor of her closet. A floor plan of the closet is shown.

a. How much carpet does Erin need?
_____ ft²

Answer: 61 ft²

Explanation:
Area of the rectangle = length x breadth = 4 x 10 = 40 ft
Area of the triangle = 1/2 x base x height = 1/2 x 6 x 7 = 21 ft
The total area of the figure = 40+21 = 61 ft²

Question 6.
b. The carpet Erin has chosen costs $2.50 per square foot. How much will it cost her to carpet the floor?
$ _____

Answer: $152.50

Explanation:
Cost per square foot of the carpet = $2.50
The total cost of the carpet on the floor = 61 x $2.50 =$152.50

9.2 Practice A Geometry Answers Question 7.
Multiple Representations
Hexagon ABCDEF has vertices A(-2, 4), B(0, 4), C(2, 1), D(5, 1), E(5, -2), and F(-2, -2). Sketch the figure on a coordinate plane. What is the area of the hexagon?

_____ units²

Answer: The area of the figure is 30 square units

Explanation:
Separate the figure into a trapezium and a rectangle.
Area of a trapezium = 1/2 (a+b) h= 1/2 (2+4) x 3 = 1/2 (6) 3 = 9 square units
Area of a rectangle = length x breadth = 7 x 3 = 21 square units
The total area of the figure = 9+21 = 30 square units

Question 8.
A field is shaped like the figure shown. What is the area of the field? Use 3.14 for π.

_____ m²

Answer: 146.24 m²

Explanation:
Divide the figure into a square, triangle and a quarter of a circle.

Area of a square = side x side = 8 x 8 = 64 m²
Area of a quarter of a circle = 1/4 (πr²) = 1/4 (3.14 x 8²)
= 1/4 (200.96) = 50.24 m²
Area of the triangle = 1/2 x base x height = 1/2 x 8 x 8 = 32 m²
Total area of the figure = 64+32+50.24 = 146.24 m²

Question 9.
A bookmark is shaped like a rectangle with a semicircle attached at both ends. The rectangle is 12 cm long and 4 cm wide. The diameter of each semicircle is the width of the rectangle. What is the area of the bookmark? Use 3.14 for π.
_____ cm²

Answer: 60.56 cm²

Explanation:
The bookmark is divided into a rectangle, a semicircle.
Area of the rectangle = length x breadth = 12 x 4 = 48 cm²
The diameter of the semicircle = The width of the rectangle = 4 cm
The radius of the semicircle = 4/2 = 2 cm
The area of the semicircle = πr² = 3.14 x 2 x 2 = 12.56 cm²
The total area of the bookmark = 12.56 + 48 = 60.56 cm²

Question 10.
Multistep
Alex is making 12 pendants for the school fair. The pattern he is using to make the pennants is shown in the figure. The fabric for the pennants costs $1.25 per square foot. How much will it cost Alex to make 12 pennants?

$ _____

Answer: $52.50

Explanation:
Each pendant is made up of a rectangle and a triangle.
Area of the rectangle = length x breadth = 3 x 1 = 3 ft²
Area of the triangle = 1/2 x base x height = 1/2 x 1 x 1 = 0.5 ft²
The total area of the pendant = 3+0.5 = 3.5 ft²
Number of pendants = 12
Area of the pendants = 12 x 3.5 = 42 ft²
Cost of each square foot of the pendant = $1.25
Total cost for all the 12 pendants = 12 x $1.25  = $52.50

Question 11.
Reasoning
A composite figure is formed by combining a square and a triangle. Its total area is 32.5 ft². The area of the triangle is 7.5 ft². What is the length of each side of the square? Explain.
_____ ft

Answer: 5 ft

Explanation:
Given:
The area of the composite figure = 32.5 ft²
The area of the triangle = 7.5 ft²
The area of the square = 32.5 – 7.5 = 25
side x side = 25
side² = 25
side = root 25 = 5 ft

H.O.T. – Page No. 282

Focus on Higher Order Thinking

Question 12.
Represent Real-World Problems
Christina plotted the shape of her garden on graph paper. She estimates that she will get about 15 carrots from each square unit. She plans to use the entire garden for carrots. About how many carrots can she expect to grow? Explain.

______ carrots

Answer: 300 carrots

Explanation:
This shape is divided into two triangles and a square.
Area of figure = 2(1/2 x 2 x 2) + 4(4) = 4 + 16 = 20 square units
Number of carrots per square unit = 300
Total number of carrots = 20 x 15 = 300

Question 13.
Analyze Relationships
The figure shown is made up of a triangle and a square. The perimeter of the figure is 56 inches. What is the area of the figure? Explain.

_____ in²

Answer: 192 in²

Explanation:
Given:
The perimeter of the figure = 56 inches
The figure is divided into a square and a triangle.
10 + 10 + 3s = 56
3s = 36
s = 12
The area of a triangle = 1/2 x 12 x 8 = 48 in²
The area of a square = 12 x 12 = 144 in²
Total area of the figure = 144 + 48 = 192 in²

Question 14.
Critical Thinking
The pattern for a scarf is shown at right. What is the area of the scarf? Use 3.14 for π.

_____ in²

Answer: 243 in²

Explanation:
Area of the rectangle in the given figure = 28 x 15 = 420 in²
Area of two semicircles = 2 (1/2 πr² ) = 3.14 x 7.5 x 7.5 = 176.625 in²
Area of the shaded region = 420 – 176.625 = 243 in²(approx)

Question 15.
Persevere in Problem-Solving
The design for the palladium window shown includes a semicircular shape at the top. The bottom is formed by squares of equal size. A shade for the window will extend 4 inches beyond the perimeter of the window, as shown by the dashed line around the window. Each square in the window has an area of 100 in².

a. What is the area of the window? Use 3.14 for π.
_____ in²

Answer: a) 2228 in²

Explanation:
Area of the square = 100 in²
side x side = 100
Side = 10 in
Since the side of each square is 10 in and there are 4 squares.
The side length of the larger square (s) = 40 in
Area of the larger square = side x side = 40 x 40 = 1600 in²
Since the side of each square is 10 in and there are 2 squares.
The radius of the semi-circle = 20 in
Area of the semi-circle = 1/2(πr²) = 1/2(3.14 x 20²) = 628 in²
The area of the window = 1600 + 628 = 2228 in²

Question 15.
b. What is the area of the shade? Round your answer to the nearest whole number.
_____ in²

Answer: b) 3016 in²

Explanation:
The shade extends 4 inches beyond the shapes so the length of the bottom rectangle is 40+4+4 = 48 in
The length extends below the original square.
The height is now = 40+4 = 44 in
The radius of the semi-circle = 20+4 = 24 in
The new area of the figure = 48(44) + 1/2(3.14 x 24²) = 2112 + 904.32 = 3016.32 = 3016 in²

Guided Practice – Page No. 286

Find the surface area of each solid figure.

Question 1.

Total surface area: _____ ft²

Answer: 150 ft²

Explanation:
The base is a triangle with side lengths of 8 ft, 5 ft, 5 ft so the perimeter of the base = P = 8+5+5 = 18 ft
The height of the prism = 7 ft
The base is a triangle.
Area of the triangle = 1/2 (8) (3) = 12 ft²
The surface area formula for a prism is S = Ph + 2b
P = Perimeter = 18 h = height = 7 b = base = area of the triangle = 12
The surface area of the prism = 18(7) + 2(12) = 126 + 24 = 150 ft²

Practice and Homework Lesson 9.2 Answer Key Question 2.

Total surface area: _____ m²

Answer: 503 m²

Explanation:
Given:
Dimensions of the cuboid:
Length = 11 m
Breadth = 9 m
Height = 7 m
The surface area of the cuboid = 2(lb+bh+hl) = 2(11 x 9 + 9 x 7 + 7 x 11) = 478m²

The dimensions of the cube:
Length of the side = 2.5 m
The surface area of the cube = 6a² = 6 x 2.5 x 2.5 = 37.5 m²
The surface area of the rectangular prism = 2.5 x 2.5 = 6.25
The surface area of the figure = The overlapping area is the area of the base of the cube
= 37.5 + 478 – 2(6.25) = 503 m²

Essential Question Check-In

Question 3.
How can you find the surface area of a composite solid made up of prisms?
Type below:
_____________

Answer: The surface area of the prisms, add them up, and then subtract the overlapping areas twice.

Explanation:
The surface area of a composite solid is made up of prisms by finding the surface areas of the prisms, adding them up, and then up, and then subtracting the overlapping areas.

Independent Practice – Page No. 287

Question 4.
Carla is wrapping a present in the box shown. How much wrapping paper does she need, not including overlap?

_____ in²

Answer: 164 in²

Explanation:
The surface area of the cuboid excluding the top = 2h(l+b) + lb = 2 x 4 ( 13 ) + 10 x 3 =  164 in²
The length of the wrapping paper = The surface area of the cuboid excluding the top = 164 in²

Question 5.
Dmitri wants to cover the top and sides of the box shown with glass tiles that are 5 mm square. How many tiles does he need?

_____ tiles

Answer: 3720 tiles

Explanation:
The surface area of the cuboid excluding the bottom = 2h(l+b) + lb = 2 x 9 (35) + 20 x 15 = 930 cm²
5mm = 0.5 cm
Area of a tile = Area of the square = a² = 0.5cm x 0.5cm = 0.25 cm²
Total number of tiles = 930/0.25 = 3720 tiles

Question 6.
Shera is building a cabinet. She is making wooden braces for the corners of the cabinet. Find the surface area of each brace.

_____ in²

Answer: 45 in²

Explanation:
The perimeter of the figure = P = 3(3) + 2(1) = 11 in
Base = B = 3(2) = 6 in
Height = h = 3
The surface area of the figure = Ph + 2B = 11 x 3 +2(6) = 33 + 12 = 45 in²

Circumference and Area Answer Key Question 7.
The doghouse shown has a floor, but no windows. Find the total surface area of the doghouse, including the door.

_____ ft²

Answer:  66ft²

Explanation:
Perimeter of the pentagon base (P) = 2(2.5) + 2(2) + 3 = 5 + 4 + 3 = 12
Area of the pentagon base by adding the area of the triangle and the area of the rectangle (B) = 1/2(3)(2) + 2(3) = 9
Height (h) = 2 + 2 = 4
The surface area of the figure = Ph + 2B = 12(4) + 2(9) = 48 + 18 = 66ft²

Eddie built the ramp to train his puppy to do tricks. Use the figure for 8–9.

Question 8.
Analyze Relationships
Describe two ways to find the surface area of the ramp.
Type below:
____________

Answer: One way is to use the formula S = Ph + 2B. Another way is to find the area of each face of the prism and add them up to get the total surface area.

Explanation:
The very first way to use the formula S = Ph + 2B is where the trapeziums are the base. The second way is to find the area of each face of the prism and then add them up to get the total surface area.

Question 9.
What is the surface area of the ramp?
_____ in²

Answer: 3264 in²

Explanation:
P = Perimeter of the figure =  16(3) + 2 (20) + 16 = 104
B = Base of the figure = 1/2 (12) (16 + 3(16)) = 6 (16 + 48) = 6 (64) = 384
h = Height of the figure = 2
Surface area of the figure = Ph + 2B = 104(2) + 2(384) = 2496 + 768 = 3264 in²

Marco and Elaine are building a stand like the one shown to display trophies. Use the figure for 10–11.

Question 10.
What is the surface area of the stand?
_____ ft²

Answer:  58 ft²

Explanation:
Top:
Perimeter = P = 4(1) = 4
Base = B = 1(1) = 1
Height = h = 3
Top surface area = Ph + 2B = 4(3) + 2(1) = 14 ft²
Bottom :
Perimeter = P = 2(7) + 2(1) = 14 + 2 = 16
Base = B = 7(1) = 7
Height = h = 2
Top surface area = Ph + 2B = 16(2) + 2(7) = 46 ft²
Overlapping area = 1(1) = 1
The surface area of the figure = The surface area of the top + The surface area of the bottom – the overlapping area = 14 + 46 – 2 = 60 – 2 = 58 ft²

Question 11.
Critique Reasoning
Marco and Elaine want to paint the entire stand silver. A can of paint covers 25 square feet and costs $6.79. They set aside $15 for paint. Is that enough? Explain.
_____

Answer: No

Explanation:
Since the surface area is 58 ft², they will need 3 cans of paint. Since each can paints 25 ft² and we cannot buy a fraction of cans.
3 cans would then cost 6.79 x 3 = 20.37 so this is not enough.

Page No. 288

Question 12.
Henry wants to cover the box shown with paper without any overlap. How many square centimeters will be covered with paper?

_____ cm²

Answer: 2316 cm²

Explanation:
Given:
Length = 24cm  Breadth = 27cm Height = 10cm
P = Perimeter = 2(24) + 2(27) = 48 + 54 = 102
B = Base = 24(27) = 648
h = Height = 10
Surface area of the figure = Ph + 2B = 102(10) + 2(648) = 1020 + 1296 = 2316 cm²

Go Math Grade 7 Lesson 9.3 Answer Key Question 13.
What If?
Suppose the length and width of the box in Exercise 12 double. Does the surface area S double? Explain.
_____

Answer: No

Explanation:
Given :
Length = 24cm x 2 = 48 cm  Breadth = 27cm x 2 = 54 cm Height = 10cm
P = 2(48) + 2(54) = 96 + 108 = 204
B = 48(54) = 2592
New Surface area = Ph + 2B = 204(10) + 2(2592) = 2040 + 5184 = 7224 cm²
Double of surface area = 2 (2316) = 4632 cm²
So the new surface area is not double of the initial area.

H.O.T.

Focus on Higher Order Thinking

Question 14.
Persevere in Problem-Solving
Enya is building a storage cupboard in the shape of a rectangular prism. The rectangular prism has a square base with side lengths of 2.5 feet and a height of 3.5 feet. Compare the amount of paint she would use to paint all but the bottom surface of the prism to the amount she would use to paint the entire prism.
Type below:
______________

Answer: The difference would just be the area in the bottom surface. It would be 6.25 ft² less.

Explanation:
The difference in the amount of paint would just be the area of the bottom surface. The area of the bottom surface is (2.5)² = 6.25.
Therefore she would paint 6.25 ft² less if she painted all but the bottom surface compared to painting the entire prism.

Question 15.
Interpret the Answer
The oatmeal box shown is shaped like a cylinder. Use a net to find the surface area S of the oatmeal box to the nearest tenth. Then find the number of square feet of cardboard needed for 1,500 oatmeal boxes. Round your answer to the nearest whole number

_____ ft²

Answer: 138.28 in² , 1440 ft²

Explanation:
Given:
Dimensions of the cylinder:
Radius: 2 in
Height: 9 in
The total surface area of the cylinder = 2πr(r+h) = 2 x 22/7 x 2 (2 + 9) = 138.28 in²

The total number of square inches needed for 1,500 oatmeal boxes = 1,500 x 138.28 = 207,300 in²
1 ft = 12 in
(1 ft)² = (12 in)²
1 ft² = 144 in²
The total number of square feet needed for 1,500 oatmeal boxes (to the nearest whole number)
= 207,300/144 = 1440 ft²

Question 16.
Analyze Relationships
A prism is made of centimeter cubes. How can you find the surface area of the prism in Figure 1 without using a net or a formula? How does the surface area change in Figures 2, 3, and 4? Explain.

Type below:
______________

Answer: The surface area for the first 3 figures is the same. The surface area for Figure 4 is greater than the surface area of figures 1 – 3.

Explanation:
The surface area of the first 3 figures is the same. The 3 new faces in Figure 2 have the same areas as the 3 visible faces that were removed when the top corner cube was removed. The surface area is then the same as it is in Figure 1. Similarly, the areas of the new visible faces in Figure 3 are equal to the areas of the visible faces removed from removing the corner cubes so the surface areas are the same as in Figure 1. The surface area for Figure 4 is greater than the surface areas of figures 1 – 3. Removing the cube removed 2 of the visible faces (one from the top and one from the front side) but added 4 visible faces so the surface area increases.

Guided Practice – Solving Volume Problems – Page No. 292

Question 1.
Find the volume of the triangular prism.

_____ ft³

Answer: 84 ft³

Explanation:
Base area of the prism = 1/2 x 8 x 3 = 12 ft²
Height of the prism = 7 ft
Volume of the prism = (12 x 7) ft³

Question 2.
Find the volume of the trapezoidal prism.

_____ m³

Answer: 330 m³

Explanation:
Base area of the prism = 1/2 x (15 + 5) x 3 = 30 m²
Height of the prism = 11 m
Volume of the prism = (30 x 11) m³ = 330 m³

Lesson 9.5 Solving Volume Problems Answer Key Question 3.
Find the volume of the composite figure.

_____ ft²

Answer: Composite figure: 360 ft³

Explanation:
The volume of the triangular prism:
The base area of the prism = 1/2 x 4 x 6 = 12 ft²
Height = 6 ft
The volume of the triangular prism = 12 x 6 = 72 ft³

The volume of the rectangular prism:
The base area of the prism = 4 x 6 = 24 ft²
Height = 12 ft
The volume of the triangular prism = 12 x 24 = 288 ft³

Volume of the composite figure = (288 + 72)ft³ = 360 ft³

Find the volume of each figure.

Question 4.
The figure shows a barn that Mr. Fowler is building for his farm.

_____ ft³

Answer: 40,000 ft³

Explanation:
Triangular prism:
B = Base area = 1/2 x 10 (40) = 200 cm²
Height = 50 cm
The volume of the triangular prism = Bh = 200 x 50 = 10,000 cm³
Rectangular prism:
B = Base area =40 x 15 = 600 cm²
Height = 50 cm
The volume of the triangular prism = Bh = 600 x 50 = 30,000 cm³
Total volume of the prism = 10,000 + 30,000 = 40,000 cm³

Question 5.
The figure shows a container, in the shape of a trapezoidal prism, that Pete filled with sand.

_____ cm³

Answer: 385 cm³

Explanation:
B = Base area = 1/2 x 5 (10 + 12) = 55 cm²
Height = 7 cm
The volume of the container = Bh = 55 x 55 = 385 cm³

Essential Question Check-In

Question 6.
How do you find the volume of a composite solid formed by two or more prisms?
Type below:
______________

Answer: Finding the volume of each figure and adding them up to get the volume of the composite solid.

Explanation:
To find the volume of the composite figure that can be divided into 2 or more prisms, find the volume of each prism and add them up to get the volume of the composite solid.

Independent Practice – Page No. 293

Question 7.
A trap for insects is in the shape of a triangular prism. The area of the base is 3.5 in² and the height of the prism is 5 in. What is the volume of this trap?
_____ in³

Answer: 17.5 in³

Explanation:
The volume of the trap = Base area x height = 3.5 x 5 = 17.5 in³

Question 8.
Arletta built a cardboard ramp for her little brothers’ toy cars. Identify the shape of the ramp. Then find its volume.

Shape: _________
Area: _________ in³

Answer: 525 in³

Explanation:
Base area = 1/2 x 6 x 25 = 75 in²
Height  = 7 in
Volume of the figure = 75 x 7 = 525 in³

Question 9.
Alex made a sketch for a homemade soccer goal he plans to build. The goal will be in the shape of a triangular prism. The legs of the right triangles at the sides of his goal measure 4 ft and 8 ft, and the opening along the front is 24 ft. How much space is contained within this goal?

_____ ft³

Answer: 384 ft³

Explanation:
Base area = 1/2 x 4 x 8 = 16 ft²
Height  = 24 ft
Volume of the figure = 16 x 24 = 384 ft³

Question 10.
A gift box is in the shape of a trapezoidal prism with base lengths of 7 inches and 5 inches and a height of 4 inches. The height of the gift box is 8 inches. What is the volume of the gift box?
_____ in³

Answer: 192 in³

Explanation:
Base area = 1/2 x 4 x (7+5) = 24 in²
Height  = 8 in
Volume of the figure = 24 x 8 = 192 Base area = 1/2 x 6 x 25 = 75 in²
Height  = 7 in
Volume of the figure = 75 x 7 = 525 in³

Question 11.
Explain the Error
A student wrote this statement: “A triangular prism has a height of 15 inches and a base area of 20 square inches. The volume of the prism is 300 square inches.” Identify and correct the error.
Type below:
____________

Answer: The error is measurement unit.

Explanation:
The volume of the prism is:
base area x height = 20 x 15 = 300 in³

Find the volume of each figure. Round to the nearest hundredth if necessary.

Question 12.

_____ in³

Answer: 97.2 in³

Explanation:
The volume of the hexagonal prism = 23.4 x  3 = 70.2 in³

Base area of the rectangular prism = 3 x 3 = 9 in²
The volume of the rectangular prism = Bh = 9 x 3 = 27 in³

Total volume of the figure = 70.2 + 27 = 97.2 in³

Question 13.

_____ m³

Answer: 316.41 m³

Explanation:
The volume of the rectangular prism on the left = Bh = [7.5 x 3.75] (3.75) = 105.47 m³
The volume of the rectangular prism on the right = Bh = [7.5 x 3.75](7.5) = 210.94 m³
Total volume of the composite figure = 105.47 + 210.94 = 316.41 m³

Question 14.
Multi-Step
Josie has 260 cubic centimeters of candle wax. She wants to make a hexagonal prism candle with a base area of 21 square centimeters and a height of 8 centimeters. She also wants to make a triangular prism candle with a height of 14 centimeters. Can the base area of the triangular prism candle be 7 square centimeters? Explain.
_____

Answer: No

Explanation:
The volume of the hexagonal prism = 21 x 8 = 168
The total volume of wax, 260 is equal to the sum of the volumes of each prism.
B is the base area of the triangular prism.
168 + 14B = 260 cm³
14B = 260 – 168
B = 6.6 cm³

Page No. 294

Question 15.
A movie theater offers popcorn in two different containers for the same price. One container is a trapezoidal prism with a base area of 36 square inches and a height of 5 inches. The other container is a triangular prism with a base area of 32 square inches and a height of 6 inches. Which container is the better deal? Explain.

Type below:
___________

Answer: The triangular prism is a better deal since it has a larger volume

Explanation:
The base area of the trapezoidal prism = 36 in²
The volume of the trapezoidal prism = Bh = 36 x 5 = 175 in³

The base area of the triangular prism = 32 in²
The volume of the rectangular prism = Bh = 32 x 6 = 192 in³

The triangular prism is a better deal since it has a larger volume.

H.O.T.

Focus on Higher Order Thinking

Question 16.
Critical Thinking
The wading pool shown is a trapezoidal prism with a total volume of 286 cubic feet. What is the missing dimension?

______ ft.

Answer: 3.5 ft

Explanation:
Area of the trapezoidal prism = B = 1/2 x 13 (2+x)
Volume of the figure = 286 cubic feet
V = Bh
286 = 1/2 x 13 (2+x)(8)
5.5 = (2+x)
x = 3.5 ft

Question 17.
Persevere in Problem-Solving
Lynette has a metal doorstop with the dimensions shown. Each cubic centimeter of the metal in the doorstop has a mass of about 8.6 grams. Find the volume of the metal in the doorstop. Then find the mass of the doorstop.

______ grams

Answer: 75 cubic centimeter, 645 grams

Explanation:
V = Bh
B = Area of the triangle of base = 10 cm , height = 6 cm = 1/2 x 10 x 6 = 30 square centimeter
V = 30 x 2.5 = 75 cubic centimeter

1 cubic centimeter = 8.6 grams in mass
V = 75 cubic centimeter x 8.6 = 645 grams

Question 18.
Analyze Relationships
What effect would tripling all the dimensions of a triangular prism have on the volume of the prism? Explain your reasoning.
Type below:
____________

Answer: The volume is 27 times the original volume.

Explanation:
The area of the base = B = 1/2 (3b) (3h) = 9/2 (bh)
H is the height of the prism
The volume would be = 9/2 (bh) x (3H) = 27 [ 1/2 (bhH) ]

Therefore, The volume is 27 times the original volume.

Question 19.
Persevere in Problem Solving
Each of two trapezoidal prisms has a volume of 120 cubic centimetres. The prisms have no dimensions in common. Give possible dimensions for each prism.
Type below:
____________

Answer: A possible combination of dimension could be the height at 8 cm, base at 2 cm and 3 cm

Explanation:
The numbers that multiply to get 120 are 20 and 6 so let the first prism have a base area of 20 square centimetres and the height of 6 cm.
If the base area is 20, the height of the trapezoid and the length of the bases could be 8,2 and 3 respectively.

The other numbers that multiply to get 120 are 4 and 30 so let the second prism have a base area of 30 square centimetres and the height of 4 cm.
If the base area is 30, the height of the trapezoid and the length of the bases could be 10,1 and 5 respectively.

9.1, 9.2 Circumference and Area of Circles – Page No. 295

Find the circumference and area of each circle. Use 3.14 for π. Round to the nearest hundredth if necessary.

Question 1.

C = _________ m
A = _________ m²

Answer:
C = 43.96 m
A = 153.86 m²

Explanation:
C = 2 πr = 2 π(7) = 14 (3.14) = 43.96 m
A = πr² = 3.14 (7)² = 153.86 m²

Chapter 9 Circumference and Area of a Circle Answer Key Question 2.

C = _________ ft
A = _________ ft²

Answer:
C = 37.68 ft
A = 113.04 ft²

Explanation:
Diameter = 12 ft
Radius = d/2 = 12/2 = 6 ft
C = 2 πr = 2 π(6) = 6 (3.14) = 37.68 ft
A = πr² = 3.14 (6)² = 113.04 ft²

9.3 Area of Composite Figures

Find the area of each figure. Use 3.14 for π.

Question 3.

______ m²

Answer: 180.48 m²

Explanation:
Area of the triangle = 1/2 x 16 x 10 = 80 m²
Area of the semicircle = 1/2 πr² = 1/2 (3.14) (8)² = 100.48 m²
The total area of the figure = 80 + 100.48 = 180.48 m²

Question 4.

______ cm²

Answer: 200 cm²

Explanation:
Area of the parallelogram = 4.5(20) = 90 cm²
Area of the rectangle = 20(5.5) = 110 cm²
The total area of the figure = 90 + 110 = 200 cm²

9.4, 9.5 Solving Surface Area and Volume Problems

Find the surface area and volume of each figure.

Question 5.

S = _________ cm²
V = _________ cm³

Answer:
S = 132 cm²
V = 60 cm³

Explanation:
Perimeter = 3+4+5 = 12 cm
Base area = Area of the triangle = 1/2 x 3 x 4 = 6
S = Ph + 2B = 12(10) + 2(6) = 120 +12 = 132 cm²

V = Bh = 6 x 10 = 60 cm³

Question 6.

S = _________ yd²
V = _________ yd³

Answer:
S = 54.5 yd²
V = 27.5 yd³

Explanation:
Perimeter = 2(2.5) + 2(2) + 4 = 13 cm
Base area = Area of the triangle + Area of the rectangle = 1/2 x 1.5 x 4 + 4(2)= 11
S = Ph + 2B = 13(2.5) + 2(11) = 32.5 +22 = 54.5 yd²

V = Bh = 11 x 2.5 = 27.5 yd³

Essential Question

Area and Circumference of a Circle Iready Quiz Question 7.
How can you use geometric figures to solve real-world problems?
Type below:
______________

Answer: We can solve real-world problems by finding surface area and volume.
Example: We can find the amount of liquid in a tank by calculating its volume.

Explanation:
Real-world problems by finding surface area and volume.
Example 1: We can find the amount of liquid in a tank by calculating its volume.
Example 2: We can find the surface area of the house and find the amount of paint required to paint the house.

Page No. 296

Question 1.
What is the circumference of the circle?

a. 34.54 m
b. 69.08 m
c. 379.94 m
d. 1519.76 m

Answer: b. 69.08 m

Explanation:
Circumference = 2 πr = 2 π(11) = 22 (3.14) = 69.08 m

Question 2.
What is the area of the circle?

Options:
a. 23.55 m²
b. 47.1 m²
c. 176.625 m²
d. 706.5 m²

Answer: c. 176.625 m²

Explanation:
Diameter = 15 m
Radius = 7.5 m
Area of the circle = πr² = 3.14 (7.5)² = 176.625 m²

Question 3.
What is the area of the figure?

Options:
a. 28.26 m²
b. 36 m²
c. 64.26 m²
d. 92.52 m²

Answer: c. 64.26 m²

Explanation:
Area of the square = 6 x 6 = 36 m²
Radius = 6 m
Area of the quarter circle = 1/4 πr² = 1/4 x 3.14 (6)² = 28.26 m²
The total area of the figure = 36 + 28.26 = 64.26 m²

Question 4.
A one-year membership to a health club costs $480. This includes a $150 fee for new members that is paid when joining. Which equation represents the monthly cost x in dollars for a new member?
Options:
a. 12x + 150 = 480
b. \(\frac{x}{12}\) + 150 = 480
c. 12x + 480 = 150
d. \(\frac{x}{12}\) + 480 = 150

Answer: a. 12x + 150 = 480

Explanation:
If x is the monthly fee, then 12x is the total monthly fee.
The joining fee = $150
Total cost = $480
then,
12x + 150 = 480

Chapter 9 Practice Test Surface Area Answer Key Question 5.
What is the volume of the prism?

Options:
a. 192 ft³
b. 48 ft³
c. 69 ft³
d. 96 ft³

Answer: d. 96 ft³

Explanation:
B = Base area of the triangle = 1/2 x 8 x 2 = 8 ft²
Height = 12 ft
Volume of the triangular orism = Bh = 8(12) = 96 ft³

Question 6.
A school snack bar sells a mix of granola and raisins. The mix includes 2 pounds of granola for every 3 pounds of raisins. How many pounds of granola are needed for a mix that includes 24 pounds of raisins?
Options:
a. 16 pounds
b. 36 pounds
c. 48 pounds
d. 120 pounds
e. 120 pounds

Answer: a. 16 pounds

Explanation:
2/3 is equal to x/24 then 3 times 8 is equal to 24 and if 2 times 8 is equal to 16.

Question 7.
Find the percent change from $20 to $25.
Options:
a. 25% decrease
b. 25% increase
c. 20% decrease
d. 20% increase

Answer: b. 25% increase

Explanation:
25 – 20 = 5 divide by 20 = 1/4
When we find the percentage we get 25.
So we can say that there is an increase in 25%

Question 8.
Each dimension of the smaller prism is half the corresponding dimension of the larger prism.

a. What is the surface area of the figure?
_____ in²

Answer: 856 in²

Explanation:
Height of the top prism = 10/2 = 5
Length of the top prism = 16/2 = 8
Width of the top prism = 8/2 = 4
Perimeter = 2l + 2w = 2(8) + 2(4) = 16 + 8 = 24 in
B = lw = 8(4) = 32 in
Surface area of top prism= Ph + 2B = 24(5) + 2(32) = 184 in²

Height of the prism = 10
Length of the prism = 16
Width of the prism = 8
Perimeter = 2l + 2w = 2(16) + 2(8) = 32 + 16 = 48 in
B = lw = 16(8) = 128 in
Surface area of bottom prism= Ph + 2B = 48(10) + 2(128) = 736 in²

Area of overlapping region = 32 in²

The total surface area of the prism
= Surface area of top prism + Surface area of bottom prism – 2[Area of overlapping region ]
= 184 + 736 – 2(32) = 856 in²

Question 8.
b. What is the volume of the figure?
_____ in³

Answer: 1440 in³

Explanation:
Volume of top prism = Bh = 32(5) = 160 in³
Volume of bottom prism = Bh = 128(10) = 1280 in³
The total volume of the figure = 160 + 1280 = 1440 in³

EXERCISES – Page No. 298

Question 1.
In the scale drawing of a park, the scale is 1 cm: 10 m. Find the area of the actual park.

_____ m²

Answer: 450 m²

Explanation:
Multiply the dimensions of the scale drawing by 10 since 1 cm = 10 m
3cm by 1.5 cm = 30m by 15 m
Area = 30(15) = 450 m²

Question 2.
Find the value of y and the measure of ∠YPS.

y = __________ °
mYPS = __________ °

Answer: y = 8
mYPS = 40 °

Explanation:
140 + 5y = 180 [sum of angle on a line = 180°]
5y = 40
y = 8

mYPS = mRPZ = 5y [vertically opposite angles]
mYPS = 5(8) = 40°

Question 3.
Kanye wants to make a triangular flower bed using logs with the lengths shown below to form the border. Can Kanye form a triangle with the logs without cutting any of them? Explain.

_____

Answer: No

Explanation:
A side of a triangle must be greater than the difference of the other two sides and smaller than the sum of the other 2 sides.
The sum of the first 2 sides = 3+4 = 7 < 8
Therefore, he cannot form a triangle unless he cuts the logs.

Circumference Homework Help Question 4.
In shop class, Adriana makes a pyramid with a 4-inch square base and a height of 6 inches. She then cuts the pyramid vertically in half as shown. What is the area of each cut surface?

_____ in²

Answer: 12 in²

Explanation:
Base = 4 in
Height = 6 in
Area of the triangle = 1/2 x 6 x 4 = 12 in²

Page No. 300

Find the circumference and area of each circle. Round to the nearest hundredth.

Question 1.

C = __________ in
A = __________ in²

Answer:
C = 69.08 in
A = 379.94 in²

Explanation:
Diameter = 22 in
Radius = d/2 = 22/2 = 11 in
C = 2 πr = 2 π(11) = 22 (3.14) = 69.08 in
A = πr² = 3.14 (11)² = 379.94 in²

Question 2.

C = __________ m
A = __________ m²

Answer:
C = 28.26 m
A = 63.59m²

Explanation:
Radius = 4.5 m
C = 2 πr = 2 π(4.5) = 9 (3.14) = 28.26 m
A = πr² = 3.14 (4.5)² = 63.59 m²

Find the area of each composite figure. Round to the nearest hundredth if necessary.

Question 3.

______ in²

Answer: 99 in²

Explanation:
Area of the square = 9 x 9 = 81 in²
Base of the triangle = 13 – 9 = 4 in
Area of the triangle = 1/2 x 4 x 9 = 18 in²
The total area of the figure = 81 + 18 = 99 in²

Question 4.

______ cm²

Answer: 420.48 cm²

Explanation:
Area of the rectangle = 16 x 20 = 320 cm²
Diameter = 16 cm
Radius = 16/2 = 8 cm
Area of the semi circle = 1/2 πr² = 1/2 x 3.14 (8)² = 100.48 cm²
The total area of the figure = 320 + 100.48 = 420.48 cm²

Find the volume of each figure.

Question 5.

______ in³

Answer: 420 in³

Explanation:
B = 7(5) = 35 in²
V = Bh = 35 x 12 = 420 in³

Question 6.
The volume of a triangular prism is 264 cubic feet. The area of a base of the prism is 48 square feet. Find the height of the prism.
______ in

Answer: 5.5 ft

Explanation:
V = Bh
264 = 48h
h = 264/48 = 5.5ft

Page No. 301

A glass paperweight has a composite shape: a square pyramid fitting exactly on top of an 8 centimeter cube. The pyramid has a height of 3 cm. Each triangular face has a height of 5 centimeters.

Question 7.
What is the volume of the paperweight?
______ cm³

Answer: 576 cm³

Explanation:
Pyramid:
B = 8 x 8 = 64 cm²
V = 1/3 Bh = 1/3 x 64 x 3 = 64 cm³
Prism:
B = 8 x 8 = 64 cm²
V = Bh = 64 x 8 = 512 cm³

The total volume of the figure = 64 + 512 = 576 cm³

Question 8.
What is the total surface area of the paperweight?
______ cm²

Answer: 400 cm²

Explanation:
Pyramid:
P = 4(8) = 32 cm
S = 1/2 Pl + B = 80 + 64 = 144 cm²

Prism:
P = 4(8) = 32 cm
S = Ph + 2B = 32(8) + 2(64) = 384 cm²
The total surface area of the prism
= Area of the prism + Area of the pyramid – 2[Area of the overlapping region]
= 144 + 384 – 2(64) = 400

Unit 4 Performance Tasks

Question 9.
Product Design Engineer
Miranda is a product design engineer working for a sporting goods company. She designs a tent in the shape of a triangular prism. The dimensions of the tent are shown in the diagram.

a. How many square feet of material does Miranda need to make the tent (including the floor)? Show your work.
______ ft²

Answer: 261 3/4  ft²

Explanation:
P = 2 x 7 1/2 + 8 = 22 1/2
B = 4/2 (8) (6) = 24
S = Ph + 2B = 22 1/2 x 9 1/2 + 2(24) = 213 3/4 + 48 = 261 3/4 ft²

Question 9.
b. What is the volume of the tent? Show your work.
______ ft³

Answer: 228 ft³

Explanation:
V = Bh = 24 x 9 1/2 = 228 ft³

Question 9.
c. Suppose Miranda wants to increase the volume of the tent by 10%. The specifications for the height (6 feet) and the width (8 feet) must stay the same. How can Miranda meet this new requirement? Explain
Type below:
____________

Answer: Increase the height to 10.45 ft

Explanation:
New volume = 1.10 x 228 = 250.8
250.8 = 24h
h = 10.45 ft

Unit 4 Performance Tasks (cont’d) – Page No. 302

Question 10.
Li is making a stand to display a sculpture made in art class. The stand will be 45 centimeters wide, 25 centimeters long, and 1.2 meters high.
a. What is the volume of the stand? Write your answer in cubic centimeters.
______ cm³

Answer: 135,000 cm³

Explanation:
B = 45 x 25 = 1125 cm²
V = Bh = 1125 x 120 = 135,000 cm³

Question 10.
b. Li needs to fill the stand with sand so that it is heavy and stable. Each piece of wood is 1 centimeter thick. The boards are put together as shown in the figure, which is not drawn to scale. How many cubic centimeters of sand does she need to fill the stand? Explain how you found your answer.
______ cm³

Answer: 116,702 cm³

Explanation:
Width = 45 – 2(1) = 43 ft
Length = 25 – 2(1) =23ft
Height = 120-2(1) = 118ft
B = 43 x 23 = 989 ft²
V = Bh = 989 x 118 = 116,702 ft³

Selected Response – Page No. 303

Question 1.
A school flag is in the shape of a rectangle with a triangle removed as shown.

What is the measure of angle x?
Options:
a. 50°
b. 80°
c. 90°
d. 100°

Answer: d. 100°

Explanation:
x = 50 + 50 = 100° [ Sum of two angles created by the 2 lines]

Circumference Area and Volume Question 2.
On a map with a scale of 2 cm = 1 km, the distance from Beau’s house to the beach is 4.6 centimetres. What is the actual distance?
Options:
a. 2.3 km
b. 4.6 km
c. 6.5 km
d. 9.2 km

Answer: a. 2.3 km

Explanation:
2/1 = 4.6/x
x = 4.6/2 = 2.3 km

Question 3.
Lalasa and Yasmin are designing a triangular banner to hang in the school gymnasium. They first draw the design on paper. The triangle has a base of 5 inches and a height of 7 inches. If 1 inch on the drawing is equivalent to 1.5 feet on the actual banner, what will the area of the actual banner be?
Options:
a. 17.5 ft²
b. 52.5 ft²
c. 39.375 ft²
d. 78.75 ft²

Answer: c. 39.375 ft²

Explanation:
1in = 1.5ft
The base of the triangle = 5 in = 1.5(5) ft = 7.5 ft
Height = 7 in = 7(1.5) ft = 10.5 ft
Area of the triangle = 1/2 x 7.5 x 10.5 = 39.375 ft²

Question 4.
Sonya has four straws of different lengths: 2 cm, 8 cm, 14 cm, and 16 cm. How many triangles can she make using the straws?
Options:
a. no triangle
b. one triangle
c. two triangles
d. more than two triangles

Answer: b. one triangle

Explanation:
The third side of a triangle must be smaller than the sum of the other two sides to form a triangle.
2+8 = 10<14
2+8 = 10<16
8+14 = 22>14
8+14 = 22>16
2+14 = 16=16
2+16 = 18>16

Therefore, only one triangle can be formed using the sides 8, 14, 16.

Question 5.
A one-topping pizza costs $15.00. Each additional topping costs $1.25. Let x be the number of additional toppings. You have $20 to spend. Which equation can you solve to find the number of additional toppings you can get on your pizza?
Options:
a. 15x + 1.25 = 20
b. 1.25x + 15 = 20
c. 15x − 1.25 = 20
d. 1.25x − 15 = 20

Answer: b. 1.25x + 15 = 20

Explanation:
If x is the number of additional toppings, then 1.25 x is the cost of the additional toppings.
This gives the total cost is 1.25x + 15
then,
1.25x + 15 = 20

Question 6.
A bank offers a home improvement loan with simple interest at an annual rate of 12%. J.T. borrows $14,000 over a period of 3 years. How much will he pay back altogether?
Options:
a. $15680
b. $17360
c. $19040
d. $20720

Answer: c. $19040

Explanation:
Simple interest = 14,000 x 0.12 x 2 = $5,040
Amount = $14,000 + $5,040 = $19040

Question 7.
What is the volume of a triangular prism that is 75 centimeters long and that has a base with an area of 30 square centimeters?
Options:
a. 2.5 cm³
b. 750 cm³
c. 1125 cm³
d. 2250 cm³

Answer: d. 2250 cm³

Explanation:
V = Bh = 30(75) = 2250cm³

Question 8.
Consider the right circular cone shown.

If a vertical plane slices through the cone to create two identical half cones, what is the shape of the cross section?
Options:
a. a rectangle
b. a square
c. a triangle
d. a circle

Answer: c. a triangle

Explanation:
Slicing through the vertex to create 2 identical half cones would create a cross-section that  is a triangle.

Page No. 304

Question 9.
The radius of the circle is given in meters. What is the circumference of the circle? Use 3.14 for π.

a. 25.12 m
b. 50.24 m
c. 200.96 m
d. 803.84 m

Answer: b. 50.24 m

Explanation:
Circumference = 2 πr = 2 π(8) = 16 (3.14) = 50.24 m

Question 10.
The dimensions of the figure are given in millimeters. What is the area of the two-dimensional figure?

Options:
a. 39 mm²
b. 169 mm²
c. 208 mm²
d. 247 mm²

Answer: c. 208 mm²

Explanation:
Area of the square = 13 x 13 = 169 mm²
Area of the triangle = 1/2 x 13 x 6 = 39 mm²
The total area of the figure = 169 + 39 = 208 mm²

Question 11.
A forest ranger wants to determine the radius of the trunk of a tree. She measures the circumference to be 8.6 feet. What is the trunk’s radius to the nearest tenth of a foot?
Options:
a. 1.4 ft
b. 2.7 ft
c. 4.3 ft
d. 17.2 ft

Answer: a. 1.4 ft

Explanation:
Circumference = 2 πr = 8.6 ft
r = 8.6/2 π = 1.4 ft

Question 12.
What is the measure in degrees of an angle that is supplementary to a 74° angle?
Options:
a. 16°
b. 74°
c. 90°
d. 106°

Answer: d. 106°

Explanation:
Sum of supplementary angles = 180°
x + 74° = 180°
x = 106°

Question 13.
What is the volume in cubic centimeters of a rectangular prism that has a length of 6.2 centimeters, a width of 3.5 centimeters, and a height of 10 centimeters?
Options:
a. 19.7 cm³
b. 108.5 cm³
c. 217 cm³
d. 237.4 cm³

Answer: c. 217 cm³

Explanation:
V = Bh
B = 6.2 x 3.5 = 21.7 cm²
h = 10 cm
V = 21.7 x 10 = 217 cm³

Question 14.
A patio is the shape of a circle with diameter shown.

What is the area of the patio? Use 3.14 for π.
Options:
a. 9 m²
b. 28.26 m²
c. 254.34 m²
d. 1017.36 m²

Answer: c. 254.34 m²

Explanation:
Diameter = 18 m
Radius = 18/2 = 9 m
Area of the patio = πr² = 3.14 (9)² = 254.34 m²

Question 15.
Petra fills a small cardboard box with sand. The dimensions of the box are 3 inches by 4 inches by 2 inches.
a. What is the volume of the box?
______ in³

Answer: 24 in³

Explanation:
V = Bh
B = 3 x 4 = 12 in²
V = 12 x 2 = 24 in³

Question 15.
b. Petra decides to cover the box by gluing on wrapping paper. How much wrapping paper does she need to cover all six sides of the box?
______ in²

Answer: 76 in²

Explanation:
P = 2(3) + 2(4) = 6 + 8 = 14 in
S = Ph + 2B = 14 x 2 + 2 x 24 = 76 in²

Question 15.
c. Petra has a second, larger box that is 6 inches by 8 inches by 4 inches. How many times larger is the volume of this second box? The surface area?
Volume is _________ times greater.
Surface area is _________ times greater

Answer: Surface area is about 2.7 times larger

Explanation:
B = 6 x 8 = 48 in²
V = Bh = 48 x 4 = 192 in³
192/24 = 8
P = 2(6) + 2(8) = 12 + 16 = 28
S = Ph + 2B = 28(4) + 2(48) = 112 + 96 = 208 in²
208/76 = 2.7

Go Math Grade 8 Answer Key Chapter 12 The Pythagorean Theorem contains 8th Standard Go Math solutions which will make students understand the concepts easily help the students to score well in the exams. This Go Math Grade 8 Answer Key Chapter 12 The Pythagorean Theorem. And in this, each and every question was explained intimately. The answers in this chapter are explained in a simple way that anyone can understand easily.

Go Math Grade 8 Answer Key Chapter 12 The Pythagorean Theorem

This chapter 12 contains Pythagorean Theorem,Converse of the Pythagorean Theorem, etc. are explained clearly which makes the scholars learn quickly. Go Math Grade 8 Answer Key Chapter 12 Chapter 12 The Pythagorean Theorem. questions are explained in a basic way that students will never feel any difficulty in learning. By this, students can gain good knowledge and this is helpful in finish student’s assignments also.

Chapter 12- Lesson 1: 

• Guided Practice – The Pythagorean Theorem – Page No. 378
• 12.1 Independent Practice – The Pythagorean Theorem – Page No. 379
• FOCUS ON HIGHER ORDER THINKING – The Pythagorean Theorem – Page No. 380

Chapter 12- Lesson 2: 

• Guided Practice – Converse of the Pythagorean Theorem – Page No. 384
• 12.2 Independent Practice – Converse of the Pythagorean Theorem – Page No. 385
• Converse of the Pythagorean Theorem – Page No. 386

Chapter 12- Lesson 3: 

• Guided Practice – Distance Between Two Points – Page No. 390
• 12.3 Independent Practice – Distance Between Two Points – Page No. 391
• Distance Between Two Points – Page No. 392
• Ready to Go On? – Model Quiz – Page No. 393
• Selected Response – Mixed Review – Page No. 394

Guided Practice – The Pythagorean Theorem – Page No. 378

Question 1.
Find the length of the missing side of the triangle

a² + b² = c² → 24² + ? = c² → ? = c²
The length of the hypotenuse is _____ feet.
_____ feet

Answer: The length of the hypotenuse is 26 feet.

Explanation: According to the Pythagorean Theorem, we shall consider values of a = 24ft, b = 10ft.
Therefore c = √(a² +b²)
c = √(24² + 10²)
= √(576 + 100)
= √676
= 26ft

The Pythagorean Theorem Answer Key Question 2.
Mr. Woo wants to ship a fishing rod that is 42 inches long to his son. He has a box with the dimensions shown.

a. Find the square of the length of the diagonal across the bottom of the box.
________ inches

Answer: 1700 inches.

Explanation: Here we consider the length of the diagonal across the bottom of the box as d.
Therefore, according to Pythagorean Theorem
W² + l² = d²
40² + 10² = d²
1600 + 100 = d²
1700 = d²

Question 2.
b. Find the length from the bottom corner to the opposite top corner to the nearest tenth. Will the fishing rod fit?
________ inches

Answer: 42.42 inches.

Explanation: We denote by r, the length from the bottom corner to the opposite top corner. We use our Pythagorean formula to find r.
h² + s² = r²
10² + 1700 = r²
100 + 1700 = r²
1800 = r²,    r = √1800 => 42.42 inches

ESSENTIAL QUESTION CHECK-IN

Question 3.
State the Pythagorean Theorem and tell how you can use it to solve problems.

Answer:
Pythagorean Theorem: In a right triangle, the sum of squares of the legs a and b is equal to the square of the hypotenuse c.
a² + b² = c²
We can use it to find the length of a side of a right triangle when the lengths of the other two sides are known.

12.1 Independent Practice – The Pythagorean Theorem – Page No. 379

Find the length of the missing side of each triangle. Round your answers to the nearest tenth.

Question 4.

________ cm

Answer: 8.9 cm.

Explanation: According to the Pythagorean theorem we consider values of a = 4cm, b = 8cm.
c² = a² + b²
= 4² + 8²
= 16 + 64
c²= 80, c= √80 => 8.944
After rounding to the nearest tenth value c= 8.9cm

Pythagorean Theorem Quiz 8th Grade Math Question 5.

________ in.

Answer: 11.5 in.

Explanation: According to the Pythagorean theorem we consider values of b = 8in, c= 14in
c² = a² + b²
14² = a² + 8²
196 = a² + 64
a² = 196 – 64
a  = √132 => 11.4891
a = 11.5 in

Question 6.
The diagonal of a rectangular big-screen TV screen measures 152 cm. The length measures 132 cm. What is the height of the screen?
________ cm

Answer: 75.4 cm

Explanation: Let’s consider the diagonal of the TV screen as C = 152cm, length as A = 132 cm, and height of the screen as B.

As C² = A² + B²
   152² = 132² + B²
23,104 = 17,424 + B²
B² = 23,104 – 17,424
B = √5680 => 75.365
So the height of the screen B = 75.4cm

Question 7.
Dylan has a square piece of metal that measures 10 inches on each side. He cuts the metal along the diagonal, forming two right triangles. What is the length of the hypotenuse of each right triangle to the nearest tenth of an inch?
________ in.

Answer: 14.1in.

Explanation:

Using the Pythagorean Theorem, we have:
a² + b² = c²
10² + 10² = c²
100 + 100 = c²
200 = c²
We are told to round the length of the hypotenuse of each right triangle to the nearest tenth of an inch, therefore: c = 14.1in

Question 8.
Represent Real-World Problems A painter has a 24-foot ladder that he is using to paint a house. For safety reasons, the ladder must be placed at least 8 feet from the base of the side of the house. To the nearest tenth of a foot, how high can the ladder safely reach?
________ ft

Answer: 22.6 ft.

Explanation: Consider the below diagram. Length of the ladder C = 24ft, placed at a distance from the base B = 8ft, let the safest height be A.

By using the Pythagorean Theorem:
C² = A² + B²
24² = A² + 8²
576 = A² + 64
A² = 576 – 64 => 512
A = √512 => 22.627
After rounding to the nearest tenth, the value of A = 22.6ft

Pythagorean Theorem Unit Test Answer Key Question 9.
What is the longest flagpole (in whole feet) that could be shipped in a box that measures 2 ft by 2 ft by 12 ft?

________ ft

Answer: The longest flagpole (in whole feet) that could be shipped in this box is 12 feet.

Explanation:
From the above diagram, we have to find the value of r, which gives us the length longest flagpole that could be shipped in the box.
Where width w = 2ft, height h = 2ft and length l = 12ft.

First find s, the length of the diagonal across the bottom of the box.
w² + l² = s²
2² + 12² = s²
4 + 144 = s²
148 = s²
We use our expression for s to find r, since triangles with sides s, r, and h also form a right-angle triangle.
h² + s² = r²
2² + 148 = r²
4 + 148 = r²
152 = r²
r = 12.33ft.

Question 10.
Sports American football fields measure 100 yards long between the end zones and are 53 \(\frac{1}{3}\) yards wide. Is the length of the diagonal across this field more or less than 120 yards? Explain.
____________

Answer: The diagonal across this field is less than 120 yards.

Explanation: From the above details we will get a diagram as shown below.

We are given l = 100 and w = 53  =  . If we denote with d the diagonal of the field, using the Pythagorean Theorem, we have:
l² + w² = d²
100² + (160/3)² = d²
10000 + (25600/9) = d²
9*10000 + 9*(25600/9) = 9* d²
90000 + 25600 = 9 d²
(115600/9) = d²
(340/9) = d²
d = 113.3
Hence the diagonal across this field is less than 120 yards.

Question 11.
Justify Reasoning A tree struck by lightning broke at a point 12 ft above the ground as shown. What was the height of the tree to the nearest tenth of a foot? Explain your reasoning.

________ ft

Answer: The total height of the tree was 52.8ft

Explanation:

By using the Pythagorean Theorem
a² + b²  = c²
12² + 39² = c²
144 + 1521 = c²
1665 = c²
We are told to round the length of the hypotenuse to the nearest tenth of a foot, therefore: c = 40.8ft.
Therefore, the total height of the tree was:
height = a+c
height = 12 +40.8
height = 52.8 feet

FOCUS ON HIGHER ORDER THINKING – The Pythagorean Theorem – Page No. 380

Question 12.
Multistep Main Street and Washington Avenue meet at a right angle. A large park begins at this corner. Joe’s school lies at the opposite corner of the park. Usually Joe walks 1.2 miles along Main Street and then 0.9 miles up Washington Avenue to get to school. Today he walked in a straight path across the park and returned home along the same path. What is the difference in distance between the two round trips? Explain.
________ mi

Answer: Joe walks 1.2 miles less if he follows the straight path across the park.

Explanation: Using the Pythagorean Theorem, we find the distance from his home to school following the straight path across the park:
a² + b²  = c²
1.2² + 0.9² = c²
1.44 + 0.81 = c²
2.25 = c²
1.5 = c
Therefore, the distance of Joe’s round trip following the path across the park is 3 miles (d_home-school + d_school-home = 1.5 + 1.5). Usually, when he walks along Main Street and Washington Avenue, the distance of his round trip is 4.2 miles (d_home-school + d_school-home = (1.2 + 0.9) + (0.9+1.2)). As we can see, Joe walks 1.2 miles less if he follows the straight path across the park.

Question 13.
Analyze Relationships An isosceles right triangle is a right triangle with congruent legs. If the length of each leg is represented by x, what algebraic expression can be used to represent the length of the hypotenuse? Explain your reasoning.

Answer: c = x√ 2

Explanation: From the Pythagorean Theorem, we know that if a and b are legs and c is the hypotenuse, then a² + b²  = c². In our case, the length of each leg is represented by x, therefore we have:
a² + b²  = c²
x² + x² = c²
2x² = c²
c = x√ 2

Pythagorean Theorem 8th Grade Math Question 14.
Persevere in Problem-Solving A square hamburger is centered on a circular bun. Both the bun and the burger have an area of 16 square inches.

a. How far, to the nearest hundredth of an inch, does each corner of the burger stick out from the bun? Explain.
________ in

Answer: Each corner of the burger sticks out 0.57 inches from the bun.

Explanation: First, we need to find the radius r of the circular bun. We know that its area A is 16 square inches, therefore:

 

A = πr²
16 = 3.14*r²
r² = (16/3.14)
r = 2.26

Then, we need to find the side s of the square hamburger. We know that its area A is 16 square inches, therefore:
A = s²
16 = s²
s = 4
Using the Pythagorean Theorem, we have to find the diagonal d of the square hamburger:
s² + s² = d²
4² + 4² = d²
16 + 16 = d²
32 = d²
d = 5.66
To find how far each corner of the burger sticks out from the bun, we denote this length by a and we get:
a = (d/2) – r => (5.66/2) – 2.26
a = 0.57.
Therefore, Each corner of the burger sticks out 0.57 inches from the bun.

Question 14.
b. How far does each bun stick out from the center of each side of the burger?
________ in

Answer: Each bun sticks out 0.26 inches from the center of each side of the burger.

Explanation:

We found that r = 2.26 and s = 4. To find how far each bun sticks out from the center of each side of the burger, we denote this length by b and we get:
b = r – (s/2) = 2.26 – (4/2)
b = 0.26 inches.

Question 14.
c. Are the distances in part a and part b equal? If not, which sticks out more, the burger or the bun? Explain.
Answer:
The distances a and b are not equal. From the calculations, we found that the burger sticks out more than the bun.

Guided Practice – Converse of the Pythagorean Theorem – Page No. 384

Question 1.
Lashandra used grid paper to construct the triangle shown.

a. What are the lengths of the sides of Lashandra’s triangle?
_______ units, _______ units, _______ units,

Answer: The length of Lashandra’s triangle is 8 units, 6 units, 10 units.

Question 1.
b. Use the converse of the Pythagorean Theorem to determine whether the triangle is a right triangle.

The triangle that Lashandra constructed is/is not a right triangle.
_______ a right triangle

Answer: Lashandra’s triangle is right angled triangle as it satisfied the Pythagorean theorem

Explanation:
Verifying with Pythagorean formula a² + b²  = c²
8^{2 +} 6² = 10²
64 + 36 =100
100 = 100.

Understand The Pythagorean Theorem Answer Key Question 2.
A triangle has side lengths of 9 cm, 12 cm, and 16 cm. Tell whether the triangle is a right triangle.
Let a = _____, b = _____, and c = ______.

By the converse of the Pythagorean Theorem, the triangle is/is not a right triangle.
_______ a right triangle

Answer: The given triangle is not a right-angled triangle

Explanation: Verifying with Pythagorean formula a² + b²  = c²
9² + 12² = 16²
81 + 144 = 256
225 ≠ 256.
Hence given dimensions are not from the right-angled triangle.

Question 3.
The marketing team at a new electronics company is designing a logo that contains a circle and a triangle. On one design, the triangle’s side lengths are 2.5 in., 6 in., and 6.5 in. Is the triangle a right triangle? Explain.
_______

Answer: It is a right-angled triangle.

Explanation: Let a = 2.5, b = 6 and c= 6.5
Verifying with Pythagorean formula a² + b²  = c²
2.5² + 6² = 6.5²
6.25 + 36 = 42.25
42.25 = 42.25.
Hence it is a right-angled triangle.

ESSENTIAL QUESTION CHECK-IN

Question 4.
How can you use the converse of the Pythagorean Theorem to tell if a triangle is a right triangle?

Answer: Knowing the side lengths, we substitute them in the formula a² + b²  = c², where c contains the biggest value. If the equation holds true, then the given triangle is a right triangle. Otherwise, it is not a right triangle.

12.2 Independent Practice – Converse of the Pythagorean Theorem – Page No. 385

Tell whether each triangle with the given side lengths is a right triangle.

Question 5.
11 cm, 60 cm, 61 cm
______________

Answer: Since 11² + 60² = 61², the triangle is a right-angled triangle.

Explanation: Let a = 11, b = 60 and c= 61
Using the converse of the Pythagorean Theorem a² + b²  = c²
11² + 60² = 61²
121 + 3600 = 3721
3721 = 3721.
Since 11² + 60² = 61², the triangle is a right-angled triangle.

Question 6.
5 ft, 12 ft, 15 ft
______________

Answer: Since 5² + 12² ≠ 15², the triangle is not a right-angled triangle.

Explanation: Let a = 5, b = 12 and c= 15
Using the converse of the Pythagorean Theorem a² + b²  = c²
 5² + 12² = 15²
25 + 144 = 225
169 ≠ 225.
Since 5² + 12² ≠ 15², the triangle is not a right-angled triangle.

Pythagorean Theorem Quick Check Answer Key Question 7.
9 in., 15 in., 17 in.
______________

Answer: Since 9² + 15² ≠ 17², the triangle is not a right-angled triangle.

Explanation: Let a = 9, b = 15 and c= 17
Using the converse of the Pythagorean Theorem a² + b²  = c²
9² + 15² = 17²
81 + 225 = 225
306 ≠ 225.
Since 9² + 15² ≠ 17², the triangle is not a right-angled triangle.

Question 8.
15 m, 36 m, 39 m
______________

Answer: Since 15² + 36² = 39², the triangle is a right-angled triangle.

Explanation: Let a = 15, b = 36 and c= 39
Using the converse of the Pythagorean Theorem a² + b²  = c²
15² + 36² = 39²
225 + 1296 = 1521
1521 = 1521.
Since 15² + 36² = 39², the triangle is a right-angled triangle.

Question 9.
20 mm, 30 mm, 40 mm
______________

Answer: Since 20² + 30² ≠ 40², the triangle is not a right-angled triangle.

Explanation: Let a = 20, b = 30 and c= 40
Using the converse of the Pythagorean Theorem a² + b²  = c²
20² + 30² = 40²
400 + 900 = 1600
1300 ≠ 1600.
Since 20² + 30² ≠ 40², the triangle is not a right-angled triangle.

Question 10.
20 cm, 48 cm, 52 cm
______________

Answer: Since 20² + 48² = 52², the triangle is a right-angled triangle.

Explanation: Let a = 20, b = 48 and c= 52
Using the converse of the Pythagorean Theorem a² + b²  = c²
20² + 48² = 52²
400 + 2304 = 2704
2704 = 2704.
Since 20² + 48² = 52², the triangle is a right-angled triangle.

Question 11.
18.5 ft, 6 ft, 17.5 ft
______________

Answer: Since 6² + 17.5² = 18.5², the triangle is a right-angled triangle.

Explanation: Let a = 6, b = 17.5 and c= 18.5
Using the converse of the Pythagorean Theorem a² + b²  = c²
6² + 17.5² = 18.5²
36 + 306.25 = 342.25
342.5 = 342.25.
Since 6² + 17.5² = 18.5², the triangle is a right-angled triangle.

Question 12.
2 mi, 1.5 mi, 2.5 mi
______________

Answer: Since 2² + 1.5² = 2.5², the triangle is a right-angled triangle.

Explanation: Let a = 2, b = 1.5 and c= 2.5
Using the converse of the Pythagorean Theorem a² + b²  = c²
 2² + 1.5² = 2.5²
4 + 2.25 = 6.25
6.25 = 6.25.
Since  2² + 1.5² = 2.5², the triangle is a right-angled triangle.

Question 13.
35 in., 45 in., 55 in.
______________

Answer: Since 35² + 45² ≠ 55², the triangle is not a right-angled triangle.

Explanation: Let a = 35, b = 45 and c= 55
Using the converse of the Pythagorean Theorem a² + b²  = c²
35² + 45² = 55²
1225 + 2025 = 3025
3250 ≠ 3025.
Since 35² + 45² ≠ 55², the triangle is not a right-angled triangle.

Question 14.
25 cm, 14 cm, 23 cm
______________

Answer: Since  14² + 23² ≠ 25², the triangle is not a right-angled triangle.

Explanation: Let a = 14, b = 23 and c= 25 (longest side)
Using the converse of the Pythagorean Theorem a² + b²  = c²
14² + 23² = 25²
196 + 529 = 625
725 ≠ 625.
Since  14² + 23² ≠25², the triangle is not a right-angled triangle.

Question 15.
The emblem on a college banner consists of the face of a tiger inside a triangle. The lengths of the sides of the triangle are 13 cm, 14 cm, and 15 cm. Is the triangle a right triangle? Explain.
________

Answer: Since  13² + 14² ≠ 15², the triangle is not a right-angled triangle.

Explanation: Let a = 13, b = 14 and c= 15
Using the converse of the Pythagorean Theorem a² + b²  = c²
13² + 14² = 15²
169 + 196 = 225
365 ≠ 225.
Since  13² + 14² ≠ 15², the triangle is not a right-angled triangle.

Question 16.
Kerry has a large triangular piece of fabric that she wants to attach to the ceiling in her bedroom. The sides of the piece of fabric measure 4.8 ft, 6.4 ft, and 8 ft. Is the fabric in the shape of a right triangle? Explain.
________

Answer: The triangular piece of fabric that Kerry has is in the shape of a right angle since it follows the Pythagorean theorem.

Explanation: Let a = 4.8, b = 6.4 and c= 8
Using the converse of the Pythagorean Theorem a² + b²  = c²
4.8² + 6.4² = 8²
23.04 + 40.96 = 64
64 = 64.
Since 4.8² + 6.4² = 8², the triangle is a right-angled triangle.

Question 17.
A mosaic consists of triangular tiles. The smallest tiles have side lengths 6 cm, 10 cm, and 12 cm. Are these tiles in the shape of right triangles? Explain.
________

Answer: Since 6² + 10² ≠ 12², by the converse of the Pythagorean Theorem, we say that the tiles are not in the shape of a right-angled triangle.

Explanation: Let a = 6, b = 10 and c= 12
Using the converse of the Pythagorean Theorem a² + b²  = c²
 6² + 10² = 12²
36 + 100 = 144
136 ≠ 144.
Since 6² + 10² ≠ 12², by the converse of the Pythagorean Theorem, we say that the tiles are not in the shape of a right-angled triangle.

The Pythagorean Theorem and Maps Answer Key Question 18.
History In ancient Egypt, surveyors made right angles by stretching a rope with evenly spaced knots as shown. Explain why the rope forms a right angle.

Answer: The rope has formed a right-angled triangle because the length of its sides follows the Pythagorean Theorem.

Explanation: The knots are evenly placed at equal distances
The lengths in terms of knots are a=4 knots, b = 3knots, c = 5 knots
Therefore a² + b²  = c²
4² + 3²  = 5²
16+9 = 25
25 = 25.
Hence rope has formed a right-angled triangle because the length of its sides follows Pythagorean Theorem.

Converse of the Pythagorean Theorem – Page No. 386

Question 19.
Justify Reasoning Yoshi has two identical triangular boards as shown. Can he use these two boards to form a rectangle? Explain.

Answer: Since it was proved that both can form a right-angled triangle, we can form a rectangle by joining them.

Explanation: Given both triangles are identical, if both are right-angled triangles then we can surely join to form a rectangle.
Let’s consider a = 0.75, b= 1 and c=1.25.
By using converse Pythagorean Theorem a² + b²  = c²
0.75² + 1² = 1.25²
0.5625 + 1 = 1.5625
1.5625 = 1.5625.
Since it was proved that both can form a right-angled triangle, we can form a rectangle by joining them.

12.2 Pythagorean Theorem Answer Key Question 20.
Critique Reasoning Shoshanna says that a triangle with side lengths 17 m, 8 m, and 15 m is not a right triangle because 17² + 8² = 353, 15² = 225, and 353 ≠ 225. Is she correct? Explain
_______

Answer: She is not right, A triangle with sides 15, 8, and 17 is a right-angled triangle.

Explanation: Lets consider a =15, b= 8 and c = 17 (which is long side)
We will verify by using converse Pythagorean Theorem a² + b²  = c²
15² + 8² = 17²
225 + 64 = 289
289 = 289.
Since the given dimensions satisfied the Pythagorean Theorem, we can say it is a right-angled triangle. In the given above statement what Shoshanna did was c² + b² = a², which is not the correct definition of the Pythagorean Theorem.

FOCUS ON HIGHER ORDER THINKING

Question 21.
Make a Conjecture Diondre says that he can take any right triangle and make a new right triangle just by doubling the side lengths. Is Diondre’s conjecture true? Test his conjecture using three different right triangles.
_______

Answer: Yes, Diondre’s conjecture is true. By doubling the sides of a right triangle would create a new right triangle.

Explanation: Given a right triangle, the Pythagorean Theorem holds. Therefore, a² + b²  = c²
If we double the side lengths of that triangle, we get:
(2a)² + (2b)²  = (2c)²
4a² + 4b² = 4c²
4(a² + b²) = 4c²
a² + b²  = c²                    
As we can see doubling the sides of a right triangle would create a new right triangle.We can test that by using three different right triangles.

The triangle with sides a = 6, b = 8, and c = 10 is a right triangle. We double its sides and check if the new triangle is a right triangle. After doubling the value of a = 12, b = 16 and c = 20.
12² + 16² = 20²
144 + 256 = 400
400 = 400
Hence proved!
Since 12² + 16² = 20², the new triangle is a right triangle by the converse of the Pythagorean Theorem.

The triangle with sides a = 3, b = 4 and c = 5 is a right triangle. We double its sides and check if the new triangle is a right triangle. After doubling the value of a = 6, b = 8 and c = 10.
6² + 8² = 10²
36 + 64 = 100
100 = 100
Hence proved!
Since 6² + 8² = 10², the new triangle is a right triangle by the converse of the Pythagorean Theorem.

The triangle with sides a = 12, b = 16 and c = 20 is a right triangle. We double its sides and check if the new triangle is a right triangle. After doubling the value of a = 24, b = 32 and c = 40.
24² + 32² = 40²
576 + 1024 = 1600
1600 = 1600
Hence proved!
Since 24² + 32² = 40², the new triangle is a right triangle by the converse of the Pythagorean Theorem.

Question 22.
Draw Conclusions A diagonal of a parallelogram measures 37 inches. The sides measure 35 inches and 1 foot. Is the parallelogram a rectangle? Explain your reasoning.
_______

Answer: Since 12² + 35² = 37², the triangle is the right triangle. Therefore, the given parallelogram is a rectangle.

Explanation: A rectangle is a parallelogram where the interior angles are right angles. To prove if the given parallelogram is a rectangle, we need to prove that the triangle formed by the diagonal of the parallelogram and two sides of it, is a right triangle. Converting all the values into inches, we have a = 12, b = 35 and c = 37. Using the converse of the Pythagorean Theorem, we have:
a² + b²  = c²
12² + 35² = 37²
144 + 1225 = 1369
1369 = 1369.
Since 12² + 35² = 37², the triangle is the right triangle. Therefore, the given parallelogram is a rectangle.

Question 23.
Represent Real-World Problems A soccer coach is marking the lines for a soccer field on a large recreation field. The dimensions of the field are to be 90 yards by 48 yards. Describe a procedure she could use to confirm that the sides of the field meet at right angles.

Answer: To confirm that the sides of the field meet at right angles, she could measure the diagonal of the field and use the converse of the Pythagorean Theorem. If a² + b²  = c² (where a = 90, b = 48, and c is the length of the diagonal), then the triangle is the right triangle. This method can be used for every corner to decide if they form right angles or not.

Guided Practice – Distance Between Two Points – Page No. 390

Question 1.
Approximate the length of the hypotenuse of the right triangle to the nearest tenth using a calculator.

_______ units

Answer: The length of the hypotenuse of the right triangle to the nearest tenth is 5.8 units.

Explanation: From the above figure let’s take
Length of the vertical leg = 3 units
Length of the horizontal leg = 5 units
let length of the hypotenuse = c
By using Pythagorean Theorem a² + b²  = c²
c² = 3² + 5²
c² = 9 +25
c = √34 => 5.830.
Therefore Length of the hypotenuse of the right triangle to the nearest tenth is 5.8 units.

Question 2.
Find the distance between the points (3, 7) and (15, 12) on the coordinate plane.
_______ units

Answer: The distance between points on the coordinate plane is 13

Explanation: So (x₁, y₁) = (3,7) and  (x₂, y₂) = (15, 12)
distance formula d = √( x₂ – x₁)² + √( y₂ – y₁)²
d = √(15 -3)² + √(12 – 7)²
d = √12² + 5²
d = √144 + 25
d = √169 => 13
Therefore distance between points on the coordinate plane is 13.

Lesson 2 Pythagorean Theorem Answer Key Question 3.
A plane leaves an airport and flies due north. Two minutes later, a second plane leaves the same airport flying due east. The flight plan shows the coordinates of the two planes 10 minutes later. The distances in the graph are measured in miles. Use the Pythagorean Theorem to find the distance shown between the two planes.

_______ miles

Answer: The distance between the two planes is 103.6 miles.

Explanation:
Length of the vertical d_v = √(80 -1)² + √(1-1)²
= √79² => 79.
Length of the horizontal d_{h =} √(68 -1)² + √(1-1)²
= √67² => 67.
Distance between the two planes D = √(79² + 67²)
= √(6241+4489) => √10730
= 103.5857 => 103.6 miles.

ESSENTIAL QUESTION CHECK-IN

Question 4.
Describe two ways to find the distance between two points on a coordinate plane.

Answer:

Explanation: We can draw a right triangle whose hypotenuse is the segment connecting the two points and then use the Pythagorean Theorem to find the length of that segment. We can also the Distance formula to find the length of that segment.

For example, plot three points; (1,2), (20,2) and (20,12)

Using the Pythagorean Theorem:

The length of the horizontal leg is the absolute value of the difference between the x-coordinates of the points (1,2) and (20,2).
|1 – 20| = 19
The length of the horizontal leg is 19.

The length of the vertical leg is the absolute value of the difference between the y-coordinates of the points (20,2) and (20,12).
|2 – 12| = 10
The length of the vertical leg is 10.

Let a = 19, b = 10 and let c represent the hypotenuse. Find c.
a² + b²  = c²
19² + 10²  = c²
361 + 100 = c²
461 = c²
distance is 21.5 = c

Using the Distance formula:
d= √( x₂ – x₁)² + √( y₂ – y₁)²
The length of the horizontal leg is between (1,2) and (20,2).
d= √( x₂ – x₁)² + √( y₂ – y₁)²
  =  √(20 -1)² + √(2-2)²
= √(19)² + √(0)²
= √361 => 19
The length of the vertical leg is between (20,2) and (20,12).
d= √( x₂ – x₁)² + √( y₂ – y₁)²
  =  √(20 -20)² + √(12-2)²
= √(0)² +√(10)²
= √100 => 10
The length of the diagonal leg is between (1,2) and (20,12).
d= √( x₂ – x₁)² + √( y₂ – y₁)²
  =  √(20 -1)² + √(12-2)²
= √(19)² + √(10)²
= √(361+100) => √461 = 21.5

12.3 Independent Practice – Distance Between Two Points – Page No. 391

Question 5.
A metal worker traced a triangular piece of sheet metal on a coordinate plane, as shown. The units represent inches. What is the length of the longest side of the metal triangle? Approximate the length to the nearest tenth of an inch using a calculator. Check that your answer is reasonable.

_______ in.

Answer: The length of the longest side of the metal triangle to the nearest tenth is 7.8 units.

Explanation: From the above figure let’s take
Length of the vertical leg = 6 units
Length of the horizontal leg = 5 units
let length of the hypotenuse = c
By using Pythagorean Theorem a² + b²  = c²
c² = 6² + 5²
c² = 36 +25
c = √61 => 7.8
Therefore Length of the longest side of the metal triangle to the nearest tenth is 7.8 units.

Question 6.
When a coordinate grid is superimposed on a map of Harrisburg, the high school is located at (17, 21) and the town park is located at (28, 13). If each unit represents 1 mile, how many miles apart are the high school and the town park? Round your answer to the nearest tenth.
_______ miles

Answer: The high school and the town park are 13.6 miles apart.

Explanation: The coordinates of the high school are said to be (17,21), whereas the coordinates of the park are (28,13). In a coordinate plane, the distance d between the points (17,21) and (28,13) is:

d= √( x₂ – x₁)² + √( y₂ – y₁)²
  =  √(28 -17)² + √(13-21)²
= √(11)² + √(-8)²
= √(121+64) => √185 = 13.6014

Rounding the answer to the nearest tenth:
d = 13.6.
Taking into consideration that each unit represents 1 mile, the high school and town park are 13.6 miles apart.

Question 7.
The coordinates of the vertices of a rectangle are given by R(- 3, – 4), E(- 3, 4), C (4, 4), and T (4, – 4). Plot these points on the coordinate plane at the right and connect them to draw the rectangle. Then connect points E and T to form diagonal \(\overline { ET } \).
a. Use the Pythagorean Theorem to find the exact length of \(\overline { ET } \).

Answer: The diagonal ET is about 10.63 units long.

Explanation:
Taking into consideration the triangle TRE, the length of the vertical leg (ER) is 8 units. The length of the horizontal leg (RT) is 7 units. Let a = 8 and b =7. Let c represent the length of the hypotenuse, the diagonal ET. We use the Pythagorean Theorem to find c.
a² + b²  = c²
c² = 8² + 7²
c² = 64 +49
c = √113 => 10,63.
The diagonal ET is about 10.63 units long.

Question 7.
b. How can you use the Distance Formula to find the length of \(\overline { ET } \) ? Show that the Distance Formula gives the same answer.

Answer: The diagonal ET is about 10.63 units long. As we can see the answer is the same as the one we found using the Pythagorean Theorem.

Explanation: Using the distance formula, in a coordinate plane, the distance d between the points E(-3,4) and T(4, -4) is:
d= √( x₂ – x₁)² + √( y₂ – y₁)²
  =  √(4 – (-3))² + √(- 4 – 4)²
= √(7)² + √(-8)²
= √(49+64) => √113 = 10.63.
The diagonal ET is about 10.63 units long. As we can see the answer is the same as the one we found using the Pythagorean Theorem.

Grade 8 Lesson 3 Distance in the Coordinate Plane Answer Key Question 8.
Multistep The locations of three ships are represented on a coordinate grid by the following points: P(-2, 5), Q(-7, -5), and R(2, -3). Which ships are farthest apart?

Answer: Ships P and Q are farthest apart

Explanation: Distance Formula: In a coordinate plane, the distance d between two points (x₁,y₁) and (x₂,y₂) is:

d= √( x₂ – x₁)² + √( y₂ – y₁)²
The distance d₁ between the two points P(-2,5) and Q(-7,-5) is:
d_{1 =} √( x_Q – x_P)² + √( y_Q – y_P)²
= √(-7 – (-2))² + √(- 5 – 5)²
= √(-5)² + √(-10)²
= √(25+100) => √125 = 11.18

The distance d₂ between the two points Q(-7,-5) and R(2,-3) is:
d_{3 =} √( x_R – x_Q)² + √( y_R – y_Q)²
  = √(2 – (-7))² + √(- 3 – 5)²
= √(9)² + √(2)²
= √(81+4) => √85 = 9.22

The distance d₃ between the two points P(-2,5) and R(2,-3) is:
d_{3 =} √( x_R – x_P)² + √( y_R – y_P)²
= √(2 – (-2))² + √(- 3 – 5)²
= √(4)² + √(-8)²
= √(16+64) => √80 = 8.94.
As we can see, the greatest distance is d₁ 11.8, which means that ships P and Q are farthest apart.

Distance Between Two Points – Page No. 392

Question 9.
Make a Conjecture Find as many points as you can that are 5 units from the origin. Make a conjecture about the shape formed if all the points 5 units from the origin were connected.

Answer: (0,5), (3,4), (4,3),(5,0),(4,-3),(3,-4),(0,-5),(-3,-4),(-4,-3),(-5,0),(-4,3),(-3,4).

Explanation: Some of the points that are 5 units away from the origin are: (0,5), (3,4), (4,3),(5,0),(4,-3),(3,-4),(0,-5),(-3,-4),(-4,-3),(-5,0),(-4,3),(-3,4) etc If all the points 5 units away from the origin are connected, a circle would be formed.

Question 10.
Justify Reasoning The graph shows the location of a motion detector that has a maximum range of 34 feet. A peacock at point P displays its tail feathers. Will the motion detector sense this motion? Explain.

Answer: Considering each unit represents 1 foot, the motion detector, and peacock are 33.5 feet apart. Since the motion detector has a maximum range of 34 feet, it means that it will sense the motion of the peacock’s feathers.

Explanation: The coordinates of the motion detector are said to be (0,25), whereas the coordinates of the peacock are (30,10). In a coordinate plane, the distance d between the points (0,25) and (30,10) is:
d ₌ √( x₂ – x₁)² + √( y₂ – y₁)²
= √(30 – 0)² + √(10 – 25)²
= √(30)² + √(-15)²
= √(900+225) => √1125.
Rounding answer to the nearest tenth:
d = 33.5 feet.
Considering each unit represents 1 foot, the motion detector and peacock are 33.5 feet apart. Since the motion detector has a maximum range of 34 feet, it means that it will sense the motion of the peacock’s feathers.

FOCUS ON HIGHER ORDER THINKING

Question 11.
Persevere in Problem-Solving One leg of an isosceles right triangle has endpoints (1, 1) and (6, 1). The other leg passes through the point (6, 2). Draw the triangle on the coordinate plane. Then show how you can use the Distance Formula to find the length of the hypotenuse. Round your answer to the nearest tenth.

Answer: 7.1 units.

Explanation:

One leg of an isosceles right triangle has endpoints (1,1) and (6,1), which means that the leg is 5 units long. Since the triangle is isosceles, the other leg should be 5 units long too, therefore the endpoints of the second leg that passes through the point (6,2) are (6,1) and (6,6).
In the coordinate plane, the length of the hypotenuse is the distance d between the points (1,1) and (6,6).
d ₌ √( x₂ – x₁)² + √( y₂ – y₁)²
= √(6 – 1)² + √(6 – 1)²
= √(5)² + √(5)²
= √(25+25) => √50.
Rounding answer to nearest tenth:
d = 7.1.
The hypotenuse is around 7.1 units long.

Question 12.
Represent Real-World Problems The figure shows a representation of a football field. The units represent yards. A sports analyst marks the locations of the football from where it was thrown (point A) and where it was caught (point B). Explain how you can use the Pythagorean Theorem to find the distance the ball was thrown. Then find the distance.

_______ yards

Answer: The distance between point A and B is 37 yards

Explanation:

To find the distance between points A and B, we draw segment AB and label its length d. Then we draw vertical segment AC and Horizontal segment CB. We label the lengths of these segments a and b. triangle ACB is a right triangle with hypotenuse AB.
Since AC is vertical segment, its length, a, is the difference between its y-coordinates. Therefore, a = 26 – 14 = 12 units.
Since CB is horizontal segment, its length b is the difference between its x-coordinates. Therefore, b = 75 – 40 = 35units.
We use the Pythagorean Theorem to find d, the length of segment AB.
d² = a² + b²
d² = 12² + 35²
d² = 144 + 1225
d² = 1369 => d = √1369 => 37
The distance between points A and B is 37 yards

Ready to Go On? – Model Quiz – Page No. 393

12.1 The Pythagorean Theorem

Find the length of the missing side.

Question 1.

________ meters

Answer: Length of missing side is 28m

Explanation: Lets consider value of a = 21 and c = 35.
Using Pythagorean Theorem a² + b²  = c²
21² + b² = 35²                                            
441 + b² = 1225
b²= 784 => b = √784 = 28.
Therefore length of missing side is 28m.

Question 2.

________ ft

Answer: Length of missing side is 34ft

Explanation: Let’s consider value of a = 16 and b = 30.
Using Pythagorean Theorem a² + b²  = c²
16² + 30² = c²                                              
256 + 900 = c²
c²= 1156 => c = √1156 = 34.
Therefore length of missing side is 34ft.

12.2 Converse of the Pythagorean Theorem

Tell whether each triangle with the given side lengths is a right triangle.

Question 3.
11, 60, 61
____________

Answer: Since 11² + 60² = 61², by the converse of the Pythagorean Theorem, we say that the given sides are in the shape of a right-angled triangle.

Explanation: Let a = 11, b = 60 and c= 61
Using the converse of the Pythagorean Theorem a² + b²  = c²
11² + 60² = 61²
121 + 3600 = 3721
3721 = 3721
Since 11² + 60² = 61², by the converse of the Pythagorean Theorem, we say that the given sides are in the shape of a right-angled triangle.                      
Question 4.
9, 37, 40
____________

Answer: Since  9² + 37² ≠ 40², by the converse of the Pythagorean Theorem, we say that the given sides are not in the shape of a right-angled triangle.

Explanation: Let a = 9, b = 37 and c= 40
Using the converse of the Pythagorean Theorem a² + b²  = c²
9² + 37² = 40²
81 + 1369 = 1600
1450 ≠ 3721.
Since  9² + 37² ≠ 40², by the converse of the Pythagorean Theorem, we say that the given sides are not in the shape of a right-angled triangle.

Pythagorean Theorem Test Grade 8 Question 5.
15, 35, 38
____________

Answer: Since 15² + 35² ≠ 38², by the converse of the Pythagorean Theorem, we say that the given sides are not in the shape of a right-angled triangle.

Explanation: Let a = 15, b = 35 and c= 38
Using the converse of the Pythagorean Theorem a² + b²  = c²
15² + 35² = 38²
225 + 1225 = 1444
1450 ≠ 1444
Since 15² + 35² ≠ 38², by the converse of the Pythagorean Theorem, we say that the given sides are not in the shape of a right-angled triangle.                                                                        

Question 6.
28, 45, 53
____________

Answer: Since 28² + 45² = 53², by the converse of the Pythagorean Theorem, we say that the given sides are in the shape of right-angled triangle.

Explanation: Let a = 28, b = 45 and c= 53
Using the converse of the Pythagorean Theorem a² + b²  = c²
28² + 45² = 53²
784 + 2025 = 2809
2809 = 2809
Since 28² + 45² = 53², by the converse of the Pythagorean Theorem, we say that the given sides are in the shape of right-angled triangle.                                
Question 7.
Keelie has a triangular-shaped card. The lengths of its sides are 4.5 cm, 6 cm, and 7.5 cm. Is the card a right triangle?
____________

Answer: Since 4.5² + 6² = 7.5², by the converse of the Pythagorean Theorem, we say that the given sides are in the shape of right-angled triangle.

Explanation: Let a = 4.5, b = 6 and c= 7.5
Using the converse of the Pythagorean Theorem a² + b²  = c²
4.5² + 6² = 7.5²
20.25 + 36 = 56.25
56.25= 56.25
Since 4.5² + 6² = 7.5², by the converse of the Pythagorean Theorem, we say that the given sides are in the shape of right-angled triangle.                                                                            

12.3 Distance Between Two Points

Find the distance between the given points. Round to the nearest tenth.

Question 8.
A and B
________ units

Answer: The distance between A and B is 6.7 units

Explanation: A= (-2,3) and B= (4,6)

Distance between A and B is d ₌ √( x₂ – x₁)² + √( y₂ – y₁)²
= √(4 – (-2)² + √(6 – 3)²
= √(6)² + √(3)²
= √(36+9) => √45 = 6.7 units

Question 9.
B and C
________ units

Answer: The distance between B and C is 7.07 units

Explanation: B= (4,6) and C= (3,1)

Distance between B and C is d ₌ √( x₂ – x₁)² + √( y₂ – y₁)²
= √(4 – 3)² + √(6 – (-1))²
= √(1)² + √(7)²
= √(1+49) => √50 = 7.07 units

Question 10.
A and C
________ units

Answer: The distance between A and C is 6.403 units

Explanation: A= (-2,3) and C= (3, -1)

Distance between A and C is d ₌ √( x₂ – x₁)² + √( y₂ – y₁)²
= √(3 – (-2)² + √(-1 – 3)²
= √(5)² + √(-4)²
= √(25+16) => √41 = 6.403 units

ESSENTIAL QUESTION

Question 11.
How can you use the Pythagorean Theorem to solve real-world problems?

Answer: We can use the Pythagorean Theorem to find the length of a side of a right triangle when we know the lengths of the other two sides. This application is usually used in architecture or other physical construction projects. For example, it can be used to find the length of a ladder, if we know the height of the wall and distance on the ground from the wall of the ladder.

Selected Response – Mixed Review – Page No. 394

Question 1.
What is the missing length of the side?

A. 9 ft
B. 30 ft
C. 39 ft
D. 120 ft

Answer: C

Explanation:
Given a= 80 ft
b= ?
c= 89 ft
As a²+b²=c ²
80²+b²= 89²
6,400+b²= 7,921
b²= 7,921-6,400
b= √1,521
b= 39 ft.

Pythagorean Theorem Test Grade 8 Pdf Question 2.
Which relation does not represent a function?
Options:
A. (0, 8), (3, 8), (1, 6)
B. (4, 2), (6, 1), (8, 9)
C. (1, 20), (2, 23), (9, 26)
D. (0, 3), (2, 3), (2, 0)

Answer: D

Explanation: The value of X is the same for 2 points and 2 values of Y [(2, 3), (2, 0)]. The value of X is repeated for a function to exist, no two points can have the same X coordinates.

Question 3.
Two sides of a right triangle have lengths of 72 cm and 97 cm. The third side is not the hypotenuse. How long is the third side?
Options:
A. 25 cm
B. 45 cm
C. 65 cm
D. 121 cm

Answer: C

Explanation:
Given a= 72 cm
b= ?
c= 97 cm
As a²+b²=c ²
72²+b²= 97²
5,184+b²= 9,409
b²= 9,409-5,184
b= √4,225
b= 65 cm.

Question 4.
To the nearest tenth, what is the distance between point F and point G?

Options:
A. 4.5 units
B. 5.0 units
C. 7.3 units
D. 20 units

Answer: A.

Explanation:
Given F= (-1,6) =(x1,y1).
G= (3,4) = (x2,y2).
The difference between F&G points is
d= √(x2-x1)² + (y2-y1)²
=  √(3 – (-1))² + (4 – 6)²
 = √(4)² + (-2)²
= √16+4
= √20
= 4.471
= 4.5 units.

Question 5.
A flagpole is 53 feet tall. A rope is tied to the top of the flagpole and secured to the ground 28 feet from the base of the flagpole. What is the length of the rope?
Options:
A. 25 feet
B. 45 feet
C. 53 feet
D. 60 feet

Answer: D

Explanation:

By Pythagorean theorem
a²+b²=c ²
53²+28²= C²
2,809+784= C²
C² = 9,409-5,184
C² = 3,593
C= √3,593
C= 59.94 feet
=60 feet.

Question 6.
Which set of lengths are not the side lengths of a right triangle?
Options:
A. 36, 77, 85
B. 20, 99, 101
C. 27, 120, 123
D. 24, 33, 42

Answer: D.

Explanation:
Check if side lengths in option A form a right triangle.
Let a= 36, b= 77, c= 85
By Pythagorean theorem
a²+b²=c ²
36²+77²= 85²
1,296+ 5,929= 7,225
7,225= 7,225
As 36²+77²= 85² the triangle is a right triangle.

Check if side lengths in option B form a right triangle.
Let a= 20, b= 99, c= 101
By Pythagorean theorem
a²+b²=c ²
20²+99²= 101²
400+ 9,801= 10,201
10,201= 10,201
As 20²+99²= 101² the triangle is a right triangle.

Check if side lengths in option B form a right triangle.
Let a= 27, b= 120, c= 123
By Pythagorean theorem
a²+b²=c ²
27²+120²= 123²
729+ 14,400= 15,129
15,129= 15,129
As 27²+120²= 123² the triangle is a right triangle.

Check if side lengths in option B form a right triangle.
Let a= 27, b= 120, c= 123
By Pythagorean theorem
a²+b²=c ²
24²+33²= 42²
576+ 1,089= 1,764.
1,665= 1,764
As 24²+33² is not equal to 42² the triangle is a right triangle.

Question 7.
A triangle has one right angle. What could the measures of the other two angles be?
Options:
A. 25° and 65°
B. 30° and 15°
C 55° and 125°
D 90° and 100°

Answer: A

Explanation:
The sum of all the angles of a triangle is 180
<A+<B+<C= 180°
<A+<B+ 90°= 180°
<A+<B= 180°-90°
<A+<B= 90, here we will verify with the given options.
25°+65°= 90°
So, the measure of the other two angles are 25° and 65°

Mini-Task

Question 8.
A fallen tree is shown on the coordinate grid below. Each unit represents 1 meter.

a. What is the distance from A to B?
_______ meters

Answer: 13.34  m.

Explanation:
A= (-5,3)
B= (8,0)
Distance between A & B is
D= √{8-(-5)² + (0-3)²
= √(13)² + (-3)²
= √169+9
= √178
= 13.34  m.

Question 8.
b. What was the height of the tree before it fell?
_______ meters

Answer: 16.3 m.

Explanation:
Length of the broken part= 13.3 m
Length of vertical part = 3 m
Total Length = 13.3 m + 3 m = 16.3 m.

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