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## Texas Go Math Grade 8 Module 8 Quiz Answer Key

**Texas Go Math Grade 8 Module 8 Ready to Go On? Answer Key**

**8.1 The Pythagorean Theorem**

**Find the length of the missing side.**

Question 1.

Answer:

Let b = 21 and c = 35 Using the Pythagorean Theorem, we have:

a^{2} + b^{2} = c^{2}

a^{2} + 21^{2} = 35^{2}

a^{2} + 441 = 1225

a^{2} + 441 – 441 = 1225 – 441

a^{2} = 784

a = \(\sqrt {281}\)

The length of the missing side is 28 m.

Question 2.

Answer:

Let a = 16 and b = 30 Using the Pythagorean Theorem, we have:

a^{2} + b^{2} = c^{2}

16^{2} + 30^{2} = c^{2}

256 + 900 = c^{2}

1156 = c^{2}

c = \(\sqrt {1156}\)

c = 34

The length of the missing side is 34 m.

**8.2 Converse of the Pythagorean Theorem**

**Tell whether each triangle with the given side lengths Is a right triangle.**

Question 3.

11, 60, 61 _____________

Answer:

Let a = 11, b = 60 and c = 61. Using the converse of the Pythagorean Theorem, we have:

a^{2} + b^{2} = c^{2}

11^{2} + 60^{2} = 61^{2}

121 + 3600 = 3721

3721 = 3721

True

Since 11^{2} + 60^{2} = 61^{2}, the triangle is a right triangle.

Question 4.

9, 37, 40 _____________

Answer:

Let a = 9, b 37 and c = 40. Using the converse of the Pythagorean Theorem, we have:

a^{2} + b^{2} = c^{2}

9^{2} + 37^{2} = 40^{2}

81 + 1369 = 1600

1450 = 1600

False

Since 9^{2} + 37^{2} ≠ 40^{2}, the triangle is not a right triangle.

Question 5.

15, 35, 38 _____________

Answer:

Let a = 15,b = 35, and c = 38.

a^{2} + b^{2} = c^{2} …………… (1)

15^{2} + 35^{2} = 38^{2} (Substitute into the formula) ……………. (2)

225 + 1225 = 1444 (Simplify) …………. (3)

1450 ≠ 1444 (Add) …………. (4)

Since 15^{2} + 35^{2} ≠ 38^{2}, the triangle is not a right triangle by the converse of the Pythagorean Theorem.

Question 6.

28, 45, 53 _____________

Answer:

Let a = 28, b = 45 and c = 53. Using the converse of the Pythagorean Theorem, we have:

a^{2} + b^{2} = c^{2}

28^{2} + 45^{2} = 53^{2}

784 + 2025 = 2809

2809 = 2809

True

Since 28^{2} + 45^{2} = 53^{2} the triangle is a right 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:

Let a = 4.5, b = 6 and c = 7.5. Using the converse of the Pythagorean Theorem, we have:

a^{2} + b^{2} = c^{2}

4.5^{2} + 6^{2} = 7.5^{2}

20.25 + 36 = 56.25

56.25 = 56.25

True

Since 4.5^{2} + 6^{2} = 7.5^{2} , the card is a right triangle.

**8.3 Distance Between Two Points**

**Find the distance between the given points. Round to the nearest tenth.**

Question 8.

A and B _____________

Answer:

Using the Distance Formula, the distance d between the points A(-2, 3) and B(4, 6) is:

d = \(\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}\)

d = \(\sqrt{(4-(-2))^{2}+(6-3)^{2}}\)

d = \(\sqrt{(6)^{2}+(3)^{2}}\)

d = \(\sqrt{36+9}\)

d = \(\sqrt{45}\)

Rounding the answer to the nearest tenth:

d ≈ 6.7

The distance between points A and B is approximately 6.7 units.

Question 9.

B and C. _____________

Answer:

Using the Distance Formula, the distance d between the points B(4, 6) and C(3, -1) is:

d = \(\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}\)

d = \(\sqrt{(3-4)^{2}+(-1-6)^{2}}\)

d = \(\sqrt{(-1)^{2}+(-7)^{2}}\)

d = \(\sqrt{1+49}\)

d = \(\sqrt{50}\)

Rounding the answer to the nearest tenth:

d ≈ 7.1

The distance between points B and C is approximately 7.1 units.

Question 10.

A and C _____________

Answer:

Using the Distance Formula, the distance d between the points A(-2, 3) and C(3, -1) is:

d = \(\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}\)

d = \(\sqrt{(3-(-2))^{2}+(-1-3)^{2}}\)

d = \(\sqrt{(5)^{2}+(-4)^{2}}\)

d = \(\sqrt{25+16}\)

d = \(\sqrt{41}\)

Rounding the answer to the nearest tenth:

d ≈ 6.4

The distance between points A and C is approximately 6.4 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 the distance on the ground from the wall of the ladder.

**Texas Go Math Grade 8 Module 8 Mixed Review Texas Test Prep Answer Key**

**Selected Response**

Question 1.

What is the missing length of the side?

(A) 9 ft

(B) 30 ft

(C) 39 ft

(D) 120 ft

Answer:

By using the Pythagorean theorem,

a² + b² = c²

80² + b² = 89²

b² = 89² – 80²

b = √89² – 80²

b = √9²

b = 9 ft

Option A is the correct answer.

Question 2.

Which relation does not represent a function?

(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 is the relation that does not represent a function.

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?

(A) 25 cm

(B) 45 cm

(C) 65 cm

(D) 121 cm

Answer:

(C) 65 cm

Explanation:

Let b = 72 and c = 97, Using the Pythagorean Theorem, we have:

a^{2} + b^{2} = c^{2}

a^{2} + 72^{2} = 97^{2}

a^{2} + 5184 = 9409

a^{2} + 5184 – 5184 = 9409 – 5184

a^{2} = 4225

a = 65

Question 4.

What is the distance between point F and point G?

(A) 4.5 units

(B) 5 units

(C) 7.3 units

(D) 20 units

Answer:

(A) 4.5 units

Explanation:

Write the coordinates of the points F(-1, 6) and G(3, 4), and find the distance between F and G. Using the Distance Formula

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?

(A) 25 feet

(B) 45 feet

(C) 53 feet

(D) 60 feet

Answer:

(D) 60 feet

Explanation:

A flagpole, rope and the length of secure point from the flagpole are forming the right triangle, where the rope is a hypotenuse Let a = 53 ft and b = 28 ft Use Pythagorean Formula to find length of the rope c.

a^{2} + b^{2} = c^{2} ……………… (1)

c^{2} = 53^{2} + 28^{2} (Substitute into formula) …………….. (2)

c^{2} = 2809 + 784 (Simplify) …………… (3)

c^{2} = 3593 (Add) …………. (4)

c = 59.94163 (Take the square root from both sides) …………. (5)

c ≈ 60 ft (Round to the nearest whole number) …………. (6)

Question 6.

Which set of lengths are not the side lengths of a right triangle?

(A) 36, 77, 85

(B) 20, 99, 101

(C) 27, 120, 123

(D) 24, 33, 42

Answer:

(D) 24, 33, 42

Explanation:

Check if side lengths in option A form a right triangle.

Let a = 36, b = 77 and c = 85. Using the converse of the Pythagorean Theorem, we have:

a^{2} + b^{2} = c^{2}

36^{2} + 77^{2} = 85^{2}

1296 + 5929 = 7225

7225 = 7225

True

Since 36^{2} + 77^{2} = 85^{2}, the triangle is a right triangle.

Check if side lengths in option B form a right triangle.

Let a = 20, b = 99 and c = 101. Using the converse of the Pythagorean Theorem, we have:

a^{2} + b^{2} = c^{2}

20^{2} + 99^{2} = 101^{2}

400 + 9801 = 10201

10201 = 10201

True

Since 20^{2} + 99^{2} = 101^{2}, the triangle is a right triangle

Check if side Lengths in option C form a right triangle

Let a = 27, b = 120 and c = 123. Using the converse of the Pythagorean Theorem, we have:

a^{2} + b^{2} = c^{2}

27^{2} + 120^{2} = 123^{2}

729 + 14400 = 15129

15129 = 15129

True

Since 27^{2} + 120^{2} = 123^{2}, the triangle is a right triangle.

Check if side lengths in option D form a right triangle

Let a = 24, b = 33 and c = 42. Using the converse of the Pythagorean Theorem, we have:

a^{2} + b^{2} = c^{2}

24^{2} + 33^{2} = 42^{2}

576 + 1089 = 1764

1665 = 1764

False

Since 24^{2} + 33^{2} ≠ 42^{2}, the triangle is not a right triangle.

Question 7.

Which is an irrational number?

(A) 5.4

(B) \(\sqrt {7}\)

(C) -13

(D) \(\frac{2}{3}\)

Answer: (D) \(\frac{2}{3}\)

Explanation:

\(\frac{2}{3}\) = 0.66666…

0.66666… is a recurring number.

So, Option D is the correct answer.

Question 8.

A triangle has one right angle. What could the measures of the other two angles be?

(A) 25° and 65°

(B) 30° and 15°

(C) 55° and 125°

(D) 90° and 100°

Answer:

A triangle has one right angle

Sum of three angles = 180°

25° + 65° + 90° = 180

So, option A is the correct answer.

**Gridded Response**

Question 9.

A right triangle has legs that measure 1.5 centimeters and 2 centimeters. What is the length of the hypotenuse in centimeters?

Answer:

A right triangle has legs that measure 1.5 centimeters and 2 centimeters.

Using the converse of the Pythagorean Theorem, we have:

a^{2} + b^{2} = c^{2}

1.5² + 2² = c²

c² = 1.5² + 2²

c² = 2.25 + 4

c² = 6.25

c = 2.5

Therefore, the length of the hypotenuse is 2.5 cm.