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)

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)

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 units. Its width is units.
Answer:

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: • = units
Width of image: • = 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: • = units
Width of image: • = 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?

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).

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.

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.

Answer:

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)

Answer:

C(-1, 1), scale factor = -2

b)

Answer:

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.

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.

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

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.

b) Scale factor 0.5

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.

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 –\(\frac{1}{2}\).

Answer:

b) Rectangle PQRS is mapped onto rectangle P’Q’R’S’ with scale factor 3

Answer:

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

Answer: C(1, 2) Scale factor = -0.5

b) Triangle PQR

Answer:

C(1, 2), Scale factor is 2.5