Math in Focus Grade 8 Chapter 8 Lesson 8.4 Answer Key Dilations

Practice the problems of Math in Focus Grade 8 Workbook Answer Key Chapter 8 Lesson 8.4 Dilations to score better marks in the exam.

Math in Focus Grade 7 Course 3 B Chapter 8 Lesson 8.4 Answer Key Dilations

Math in Focus Grade 8 Chapter 8 Lesson 8.4 Guided Practice Answer Key

Solve.

Question 1.
Which triangles are dilations of one another? Explain.

a)
Math in Focus Grade 8 Chapter 8 Lesson 8.4 Answer Key Dilations 1
Answer:
Δ STU and Δ PTR. They have a center of dilation T and the sides of Δ PTR are twice as long as the sides of Δ STU.

b)
Math in Focus Grade 8 Chapter 8 Lesson 8.4 Answer Key Dilations 2
Answer:
Δ PQS and Δ PUT. They have a center of dilation, P, and the sides of Δ PUT are 1 1/2 times as long as the sides of Δ PQS.

Copy and complete.

Question 2.
A rectangle has coordinates A (5, 1), B (3, 1), C (3, 4), and D (5, 4).

a) Find the length and width of ABCD.
The length of ABCD is Math in Focus Grade 8 Chapter 8 Lesson 8.4 Answer Key Dilations 3 units. Its width is Math in Focus Grade 8 Chapter 8 Lesson 8.4 Answer Key Dilations 3 units.
Answer:
Math in Focus Grade 8 Chapter 8 Lesson 8.4 Guided Practice Answer Key_2
The length of ABCD is 3 units. Its width is 2 units.

b) Find the length and width of the image of ABCD when dilated with scale factor 2.
Length of image: Math in Focus Grade 8 Chapter 8 Lesson 8.4 Answer Key Dilations 3Math in Focus Grade 8 Chapter 8 Lesson 8.4 Answer Key Dilations 3 = Math in Focus Grade 8 Chapter 8 Lesson 8.4 Answer Key Dilations 3 units
Width of image: Math in Focus Grade 8 Chapter 8 Lesson 8.4 Answer Key Dilations 3Math in Focus Grade 8 Chapter 8 Lesson 8.4 Answer Key Dilations 3 = Math in Focus Grade 8 Chapter 8 Lesson 8.4 Answer Key Dilations 3 units
Answer:
scale factor = 2
Length of image: 3 • 2 = 6 units
Width of image: 2 • 2 = 4 units

c) Find the length and width of the image of ABCD when dilated with scale factor \(\frac{1}{2}\).
Length of image: Math in Focus Grade 8 Chapter 8 Lesson 8.4 Answer Key Dilations 3Math in Focus Grade 8 Chapter 8 Lesson 8.4 Answer Key Dilations 3 = Math in Focus Grade 8 Chapter 8 Lesson 8.4 Answer Key Dilations 3 units
Width of image: Math in Focus Grade 8 Chapter 8 Lesson 8.4 Answer Key Dilations 3Math in Focus Grade 8 Chapter 8 Lesson 8.4 Answer Key Dilations 3 = Math in Focus Grade 8 Chapter 8 Lesson 8.4 Answer Key Dilations 3 units
Answer:
Scale factor \(\frac{1}{2}\).
Length of image: 3 • \(\frac{1}{2}\) = 3\(\frac{1}{2}\). units
Width of image: 2 • \(\frac{1}{2}\). = 1 units

d)
What are the coordinates of the image rectangle under each dilation if the center of dilation is at the origin?
Math in Focus Grade 8 Chapter 8 Lesson 8.4 Answer Key Dilations 4
Answer:
Scale factor 2:
A (5, 1) = 5 × 2 , 1 × 2 = (10, 2)
B (3, 1) = 3 × 2, 1 × 2 = (6, 1)
C (3, 4) = 3 × 2, 4 × 2 = (6, 8)
D (5, 4) = 5 × 2, 4 × 2 = (10, 8)
Scale factor 1/2:
A (5, 1) = 5 × 1/2 , 1 × 1/2 = (2.5, 0.5)
B (3, 1) = 3 × 1/2, 1 × 1/2 = (1.5, 0.5)
C (3, 4) = 3 × 1/2, 4 × 1/2 = (1.5, 2)
D (5, 4) = 5 × 1/2, 4 × 1/2 = (2.5, 2)

You may want to draw the rectangle and its images on the coordinate plane to solve c).
Math in Focus Grade 8 Chapter 8 Lesson 8.4 Answer Key Dilations 5

Technology Activity

Materials:

  • geometry software

Explore The Properties Of Dilations With Geometry Software

Step 1.
Draw a line segment AB using a geometry software program.

Step 2.
Select the Dilate function, within the Transform menu. Enter the scale factor 2 to dilate the line segment about the origin. Record your results in a table of coordinates.
Math in Focus Grade 8 Chapter 8 Lesson 8.4 Answer Key Dilations 6

Step 3.
Describe how the image of \(\overline{A B}\) is related to \(\overline{A B}\).

Step 4.
Repeat Step 1 to Step 3 using a rectangle as the original figure. For Step 2, enter the scale factor \(\frac{1}{2}\).

Step 5.
Repeat Step 1 to Step 3 using a rectangle as the original figure. For Step 2, enter the scale factor -2.

Math Journal Observe any changes in the size or shape of the figure after the dilation. Which of these properties does a dilation preserve: lengths, shape, parallel lines, or perpendicular lines? Explain.

Copy and complete on graph paper.

Question 3.
The management of a swimming pool built a springboard above the pool. The height of the springboard is a dilation of the depth of the pool with center at the origin, O and scale factor –\(\frac{1}{3}\). The depth of the pool is 4.5 meters, represented by \(\overline{S T}\) on the coordinate plane. The floor is represented by the positive x-axis and the surface of the water is represented by the negative x-axis. Draw the location and height of the stand for the springboard, \(\overline{U V}\), on a copy of this vertical cross section of the pool.
Math in Focus Grade 8 Chapter 8 Lesson 8.4 Answer Key Dilations 7
Answer:
Math in Focus Grade 8 Chapter 8 Lesson 8.4 Guided Practice Answer Key_3

Use graph paper. Use 1 grid square on both axes to represent 1 unit for the interval from -7 to 4.

Question 4.
The triangles are each mapped onto their images by a dilation. Draw each triangle and its image on a coordinate plane. Then mark and label C as the center of dilation. Find the scale factor for each triangle.
a)
Math in Focus Grade 8 Chapter 8 Lesson 8.4 Answer Key Dilations 8
Answer:
Math in Focus Grade 8 Chapter 8 Lesson 8.4 Guided Practice Answer Key_4a
C(-1, 1), scale factor = -2

b)
Math in Focus Grade 8 Chapter 8 Lesson 8.4 Answer Key Dilations 9
Answer:
Math in Focus Grade 8 Chapter 8 Lesson 8.4 Guided Practice Answer Key_4b
C(3, 4) and scale factor = 3

Math in Focus Course 3B Practice 8.4 Answer Key

Tell whether each transformation is a dilation. Explain.

Question 1.
Math in Focus Grade 8 Chapter 8 Lesson 8.4 Answer Key Dilations 10
Answer:
Yes, Δ ABC and Δ ADE have center of dilation at A and the sides of Δ ADE are twice as long as the sides of Δ ABC.

Question 2.
Math in Focus Grade 8 Chapter 8 Lesson 8.4 Answer Key Dilations 11
Answer: No it is not a dilation

Solve. Show your work.

Question 3.
Nikita wants to make a mosaic for a T-shirt’s design. She makes some dilated copies of a drawing with a photocopier. The drawing is 6 inches by 4 inches. Find the length and width of each copy with the scale factor given in a) to d). State whether each copy is an enlargement or reduction of the drawing.
a) 1.5
Answer:
Scale factor = 1.5
(6, 4) = 6 × 3/2, 4 × 3/2 = 9/2 in, 6 in, Enlargement

b) 2
Answer:
Scale factor = 2
(6, 4) = 6 × 2, 4 × 2 = (12, 8), Enlargement

c) \(\frac{1}{4}\)
Answer:
Scale factor = \(\frac{1}{4}\)
(6, 4) = 6 × \(\frac{1}{4}\), 4 × \(\frac{1}{4}\) = 1.5 in, 1 in, Reduction

d) 140%
Answer:
Scale factor = 140% = 1.4
(6, 4) = 6 × 1.4, 4 × 1.4 = 8.4 in, 5.6 in. Enlargment

Copy and complete on graph paper.

Question 4.
Timothy uses a lens to view a 2-inch pencil that is represented by \(\overline{\mathrm{AB}}\) on the coordinate plane. \(\overline{\mathrm{AB}}\) is mapped onto \(\overline{A^{\prime} B^{\prime}}\) by a dilation with center at the origin, O. Draw each image for the given scale factor.

a) Scale factor -0.5
Math in Focus Grade 8 Chapter 8 Lesson 8.4 Answer Key Dilations 12
Answer:
Observe that the center of dilation is at the origin then the method of multiplying the coordinates of \(\overline{A B}\) to the scale factor can be used. Let j be the coordinates after the dilation of A whose coordinate is (3, 2).
j = A ·  (-0.5)    Formula.
= (3, 2) · (-0.5)    Substitute the Coordinates of A.
= (-1.5, -1)    Multiply.
Thus, point A’ is on (-1.5, -1). Next, Let k be the coordinates after the dilation of B whose coordinate is (3,0).
k = B · (-0.5)    Formula.
= (3, 0) · (-0.5)   Substitute the Coordinates of B.
= (-1.5. 0)    Multtiply.
Thus, point B’ is on (-1.5, 0). Therefore, the points of \(\overline{A^{\prime} B^{\prime}}\) are on (-1.5, -1) and (-1.5, 0). The illustration for the image of AB would be as depicted below.
Math in Focus Grade 8 Chapter 8 Lesson 8.4 Answer Key Dilations A 1

b) Scale factor 0.5
Math in Focus Grade 8 Chapter 8 Lesson 8.4 Answer Key Dilations 13
Answer:
Since the center of dilation is at the origin then the method of multiplying the coordinates of \(\overline{A B}\) to the scale factor can be used. Let l be the coordinates after the dilation of A whose coordinate is (2, 3).
l =  A · 0.5    Formula.
= (2, 3) · 0.5    Substitute the Coordinates of A.
= (1, 1.5) Multiply.
Thus, point A’ is on (1, 1.5). Next, Let m be the coordinates after the dilation of B whose coordinate is (2, 1).
m = B · 0.5    Formula.
= (2. 1) · 0.5    Substitute the Coordinates of B.
= (1, 0.5)    Multiply.
Thus, point B’ is on (1, 0.5). Therefore, the points of \(\overline{A^{\prime} B^{\prime}}\) are on (1, 1.5) and (1, 0.5). The illustration for the image of \(\overline{A B}\) wouLd be as depicted below.
Math in Focus Grade 8 Chapter 8 Lesson 8.4 Answer Key Dilations A 2

Question 5.
Each figure is each mapped onto its image by a dilation with its center at the origin, O. On a copy of the coordinate plane, draw each image.

a) Triangle LMN is mapped onto triangle L’M’N’ with scale factor –\(\frac{1}{2}\).
Math in Focus Grade 8 Chapter 8 Lesson 8.4 Answer Key Dilations 14
Answer:
Math-in-Focus-Grade-8-Chapter-8-Lesson-8.4-Answer-Key-Dilations-14

b) Rectangle PQRS is mapped onto rectangle P’Q’R’S’ with scale factor 3
Math in Focus Grade 8 Chapter 8 Lesson 8.4 Answer Key Dilations 15
Answer:
Math-in-Focus-Grade-8-Chapter-8-Lesson-8.4-Answer-Key-Dilations-15

Solve on graph paper. Show your work.

Question 6.
In a room, a flashlight is used to illuminate objects and cast their shadows on a wall. Each shadow is a dilation of the object’s profile.
a) A pen at point (4, 2) is mapped onto its shadow at (6, 3) with the origin as the center of dilation. Find the scale factor.
Answer:
To find the factor of the dilation, solve for the ratio of the dilated points to the originaL points. Take the x-coordinates of the two points. The x- coordinate of the shadow is 6 and x- coordinate of the original object is 4, then
Scale Factor = \(\frac{x-\text { coordinate of Dilated Image }}{x-\text { coordinate of the Original Point }}\) Formula.
= \(\frac{6}{4}\)   Substitution.
= \(\frac{3}{2}\)    Simplify.
Therefore, the scale factor is \(\frac{3}{2}\). If the y-coordinates of the two points is the chosen one to find the scale factor, then the y- coordinate of the shadow is 3 and y- coordinate of the original object is 2, so
Scale Factor = \(\frac{y-\text { coordinate of Dilated Image }}{y-\text { coordinate of the Original Point }}\) Formula.
= \(\frac{3}{2}\)   Substitution.
Therefore, the scale factor is \(\frac{3}{2}\).

b) The shadow of a circular disc has its center at (6, 2) and radius 3 units. The circular disc has center at (2, 2) and radius 1 unit. Find the center of dilation.
Answer:
To identify the center of dilation, let x1 be the x-coordinate of the center of dilation and y2 for the y-coordinate. Then, solving the x-coordinate of the center of dilation is given by the scale factor, f multiplied by the x-coordinate of the original point denoted by x2 minus x-coordinate of the dilated image denoted by x3 divided by scale factor minus 1.
x1 = \(\frac{f x_{2}-x_{3}}{f-1}\)    Formula.
= \(\frac{3(2)-6}{3-1}\)    Substitution.
= \(\frac{6-6}{3-1}\)    Multiply.
= \(\frac{0}{2}\)   Subtract.
= 0    Simplify.
Thus, the center of dilation is on 0 of x Next, solving the y-coordinate of the center of dilation is given by the scale factor, f multiplied by the y-coordinate of the original point denoted by y2 minus y-coordinate of the dilated image denoted by y3 divided by scale factor minus 1.
y1   = \(\frac{f y_{2}-y_{3}}{f-1}\)    Formula.
= \(\frac{3(2)-2}{3-1}\)    Substitution.
= \(\frac{6-2}{3-1}\)    Multiply.
= \(\frac{4}{2}\)   Subtract.
= 2    Simplify.
Thus, the center of dilation is on 2 of y. Therefore, the center of dilation lies on point (0, 2).

Question 7.
Each figure is mapped into its image by a dilation. Draw each figure and its image on the coordinate plane. Then mark and label C as the center of dilation. Find the scale factor for each figure. Use 1 grid square on both axes to represent 1 unit for the interval from —8 to 6.

a) Quadrilateral WXYZ
Math in Focus Grade 8 Chapter 8 Lesson 8.4 Answer Key Dilations 16
Answer: C(1, 2) Scale factor = -0.5
Math in Focus Course 3B Practice 8.4 Answer Key_7a

b) Triangle PQR
Math in Focus Grade 8 Chapter 8 Lesson 8.4 Answer Key Dilations 17
Answer:
Math in Focus Course 3B Practice 8.4 Answer Key_7b
C(1, 2), Scale factor is 2.5

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