Refer to our Texas Go Math Grade 8 Answer Key Pdf to score good marks in the exams. Test yourself by practicing the problems from Texas Go Math Grade 8 Module 13 Quiz Answer Key.

13.1 Properties of Dilations

Determine whether one figure is a dilation of the other. Justify your answer.

Question 1.
Triangle XYZ has angles measuring 54° and 29°. Triangle XYZ’ has angles measuring 29° and 92°.
We need to see if triangle X’Y’Z’ is dilation of triangle XYZ Triangles are similar if alt corresponding angles are equal.

Triangle XYZ has angLes measuring 54° and 29°. Let’s calculate the measure of third angle:
180° – (54° + 29°) = 180° – 83° (Calculate) ………………. (1)
= 97° (Simplify) ………………….. (2)

Triangle X’Y’Z’ has angles measuring 29° and 92°. The measure of third angle is
180° – (29° + 92°) = 180° – 121° (Calculate) …………….. (3)
= 69° (Simplify) ……………… (4)
Since this triangles have only one pair of congruent angles, the triangle X’Y’Z’ is not a dilation of XYZ.

Question 2.
Quadrilateral DEFG has sides measuring 16 m, 28m, 24m, and 20m. Quadrilateral D’E’F’G’ has sides measuring 20 m, 35 m, 30 m, and 25 m.
Quadrilateral DEFG has sides measuring 16 m, 28 m, 24 m, and 20 m.
Quadrilateral D’E’F’G’ has sides measuring 20 m, 35 m, 30 m, and 25 m
Lets find the scale factor of the corresponding sides
$$\frac{20}{16}=\frac{35}{28}=\frac{30}{24}=\frac{25}{20}=\frac{5}{4}$$
Since the scale factor is the same, the quadrilateral D’E’F’G’ is a dilation of quadrilateral DEFG.

13.2 Algebraic Representations of Dilations

Dilate each figure with the origin as the center of dilation.

Question 3.
(x, y) → (0.8x, 0.8y)

Dilatation is (x, y) → (0.8x, 0.8y)
(0, -5) → (0 ∙ 0.8, -5 ∙ 0.8) → (0, -4) …………… (1)
(5, 0) → (5 ∙ 0.8, 2 ∙ 0.8) → (4, 0) ……………. (2)
(0, 5) → (0 ∙ 0.8, 5 ∙ 0.8) → (0, 4) ……………. (3)
(-5, 0) → (- 5 ∙ 0.8, 0 ∙ 0.8) → ( 4, 0) …………… (4)
Coordinates of vertices of the dilated figure are (0, -4), (4, 0), (0, 4) and (-4, 0)

Question 4.
(x, y) → (2.5x, 2.5y)

Dilation equation (x, y) → (2.5x, 2.5y)
(2, -1) → (2 ∙ 2.5, -1 ∙ 2.5) → (5, -2.5) ………………. (1)
(2, 2) → (∙ 2.5, 2 ∙ 2.5) → (5, 5) ………… (2)
(1, 1) → (1 ∙ 2.5, 1 ∙ 2.5) → (2.5, 2.5) ……………. (3)
Coordinates of dilated vertices are (5, 2.5), (5, 5), and (2.5, 2.5).

13.3 Dilations and Measurement

Question 5.
A rectangle with length 8 cm and width 5 cm is dilated by a scale factor of 3. What are the perimeter and area of the image?
Given,
A rectangle with a length of 8 cm and a width of 5 cm is dilated by a scale factor of 3.
length = 8 cm
width = 5 cm
We know that,
Area of the rectangle = length × width
A = 8 × 5
A = 40 sq. cm
The scale factor is 3
Area of the rectangle with scale factor = 40 × 3 = 120 sq. cm
We know that,
Perimeter of the rectangle = 2L + 2W
P = (2 × 8 + 2 × 5)
P = 16 + 10
P = 26
The scale factor is 3
The perimeter of the rectangle with a scale factor of 3 is 26 × 3 = 78 cm.

Essential Question

Question 6.
How can you use dilations to solve real-world problems?
A dilations in real life are used for making models of buildings in architecture, projects, making maps.

Texas Go Math Grade 8 Module 13 Mixed Review Texas Test Prep Answer Key

Selected Response

Question 1.
Quadrilateral HIJK has sides measuring 12 cm, 26 cm, 14 cm, and 30 cm. Which could be the side lengths of a dilation of HIJK?
(A) 24 cm, 50 cm, 28 cm, 60 cm
(B) 6 cm, 15 cm, 7 cm, 15 cm
(C) 18 cm, 39 cm, 21 cm, 45 cm
(D) 30 cm, 78 cm, 35 cm, 75 cm
Answer: (C) 18 cm, 39 cm, 21 cm, 45 cm
Explanation:
The quadrilateral HIJK has sides measuring 12 cm, 26 cm, 14 cm, and 30 cm.
Scale factor = 1.5
12 × 1.5 = 18 cm
26 × 1.5 = 39 cm
14 × 1.5 = 21 cm
30 × 1.5 = 45 cm
Thus the side lengths of dilation of HIJK are 18 cm, 39 cm, 21 cm, 45 cm
So, the correct answer is option C.

Question 2.
A rectangle has vertices (6, 4), (2, 4), (6, -2), and (2, -2). What are the coordinates of the vertices of the image after a dilation with the origin as its center and a scale factor of 2.5?
(A) (9, 6), (3, 6), (9, -3), (3, -3)
(B) (3, 2), (1, 2), (3, -1), (1, -1)
(C) (12, 8), (4, 8), (12, -4), (4, -4)
(D) (15, 10), (5, 10), (15, -5), (5, -5)
(A) (9, 6), (3, 6), (9, -3), (3, -3)

Explanation:
Let’s find coordinates of vertices after a dilation, by multiplying each coordinate by 1.5, and we can easily find the correct answer.

Coordinates of vertices of the image after a dilation are:
(6, 4) → (6 ∙ 1.5, 4 ∙ 1.5) → (9, 6) …………… (1)
(2, 4) → (2 ∙ 1.5, 4 ∙ 1.5) → (3, 6) ………….. (2)
(6, -2) → (6. 1.5, -2 ∙ 1.5) → (9, -3) ……………….. (3)
(2, 2) → (2 ∙ 1.5, 2 ∙ 1.5) → (3, 3) …………… (4)
So, the correct answer is (A)

Question 3.
Which represents the dilation shown where the black figure is the preimage?

(A) (x, y) → (1 .5x, 1 .5y)
(B) (x, y) → (2.5x, 2.5y)
(C) (x, y) → (3x, 3y)
(D) (x, y) → (6x, 6y)
(B) (x, y) → (2.5x, 2.5y)

Explanation:
First, we can see that one shape is dilation of other. Lets look at the picture and see which sides can we find measures. It’s easiest to count units on sides parallel with axis. Count, and write the measures.

Use that measures to calculate scale factor
$$\frac{5}{2}$$ = 2.5 (horizontal sides) …………… (1)
$$\frac{10}{4}$$ = 2.5 (vertical sides) …………… (2)
Scale factor is 2.5
A dilation with scale factor 2.5 is (x, y) → (2.5x, 2.5y)

Question 4.
Solve -a + 7 = 2a – 8.
(A) a = – 3
(B) a = –$$\frac{1}{3}$$
(C) a = 5
(D) a = 15
(C) a = 5

Explanation:
Lets solve equation a + 7 = 2a – 8 and than we can choose correct answer.

Given equation
-a + 7 = 2a – 8 (Rewrite) ……………… (1)
-a + 7 – 7 = 2a – 8 – 7 (Take 7 from both sides) …………….. (2)
-a = 2a – 15 (Simplify) ………….. (3)
-a – 2a = -2a – 15 – 2a (Take 2a from both sides) …………….. (4)
-3a = -15 (Simplify) …………… (5)
a = $$\frac{-15}{-3}$$ (divide both sides by -3) …………….. (6)
a = 5 (Calculate) ………………… (7)

Question 5.
An equilateral triangle has a perimeter of 24 centimeters. If the triangle is dilated by a factor of 0.5, what is the length of each side of the new triangle?
(A) 4 centimeters
(B) 12 centimeters
(C) 16 centimeters
(D) 48 centimeters
Explanation:
Given,
An equilateral triangle has a perimeter of 24 centimeters.
24/3 = 8
The triangle is dilated by a factor of 0.5 and the triangle is equilateral, the side lengths of the new triangle is
8 × 0.5 = 4
The correct answer is option A.

Question 6.
Which equation does not represent a line with an x-intercept of 3?
(A) y = -2x + 6
(B) y = –$$\frac{1}{3}$$x + 1
(C) y = $$\frac{2}{3}$$x – 2
(D) y = 3x – 1
(D) y = 3x – 1

Explanation:
Coordinates of the point where the Line intersects the x-axis are (x, 0). So, if an x-intercept is 3, then the coordinates (3, 0) satisfy the equation.

(A): y = 2x + 6 → y = 0, x = 3:
0 = -2 ∙ 3 + 6
0 = -6 + 6 (Calculate)
0 = 0 (Simplify)
= This equation represent a line with an x-intercept of 3

(B): y = –$$\frac{1}{3}$$x + 1
0 = –$$\frac{1}{3}$$ ∙ 3 + 1
0 = -1 + 1 (Calculate
0 = 0 (Simplify)
⇒ This equation represent a line with an x-intercept of 3

(C): y = $$\frac{2}{3}$$x – 2
0 = $$\frac{2}{3}$$ ∙ 3 – 2
0 = 2 – 2 (Calculate)
0 = 0 (Simplify)
= This equation represent a line with an x-intercept of 3

(D): y = 3x – 1 → y = 0, x = 3
0 = 3 ∙ 3 – 1
0 = 9 – 1 (Calculate)
0 ≠ 8 (Simplify)
= This equation does not represent a line with an x-intercept of 3

Gridded Response

Question 7.
A car is traveling at a constant speed. After 2.5 hours, the car has traveled 80 miles. If the car continues to travel at the same constant speed, how many hours will it take to travel 270 miles?