B'A'

D' C'

A

D C

B

?

EXPLORE ACTIVITY

Exploring Dilations and MeasurementThe blue rectangle is a dilation (enlargement) of the green rectangle.

Using a centimeter ruler, measure and record the length of

each side of both rectangles. Then calculate the ratios of all

pairs of corresponding sides.

AB = BC = CD = DA =

A'B' = B'C' = C'D' = D'A' =

A'B' ___ AB = B'C'

___ BC = C'D' ___ CD =

D'A' ___ DA =

What is true about the ratios that you calculated?

What scale factor was used to dilate the green rectangle

to the blue rectangle?

How are the side lengths of the blue rectangle related to the

side lengths of the green rectangle?

What is the perimeter of the green rectangle?

What is the perimeter of the blue rectangle?

How is the perimeter of the blue rectangle related to the perimeter

of the green rectangle?

A

B

L E S S O N

13.3Dilations and Measurement

ESSENTIAL QUESTIONHow do you describe the effects of dilation on linear and area measurements?

8.10.D

Two-dimensional shapes—8.10.D Model the effect on linear and area measurements of dilated two-dimensional shapes. Also 8.3.B, 8.10.A, 8.10.B

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What is the area of the green rectangle?

What is the area of the blue rectangle?

How is the area of the blue rectangle related to the area of the green

rectangle?

Reflect 1. Make a Conjecture The perimeter and area of two shapes before and

after dilation are given. How are the perimeter and area of a dilated

figure related to the perimeter and area of the original figure?

Scale factor = 2 Scale factor = 1 _ 6

C

EXPLORE ACTIVITY (cont’d)

Problem-Solving Application Understanding how dilations affect the linear and area measurements of

shapes will enable you to solve many real-world problems.

A souvenir shop sells standard-sized decks of cards

and mini-decks of cards. A card in the standard deck

is a rectangle that has a length of 3.5 inches and a width

of 2.5 inches. The perimeter of a card in the mini-deck is

6 inches. What is the area of a card in the mini-deck?

Analyze Information

I need to find the area of a mini-card. I know the

length and width of a standard card and the perimeter

of a mini-card.

EXAMPLE 1 ProblemSolving

Perimeter Area

Original 8 4

Dilation 16 16

Perimeter Area

Original 30 54

Dilation 5 1.5

Animated Math

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8.3.B

Unit 5376

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My NotesFormulate a Plan

Since the mini-card is a dilation of the standard card, the figures are

similar. Find the perimeter of the standard card, and use that to find the

scale factor. Then use the scale factor to find the area of the mini-card.

Justify and EvaluateSolve

Find the perimeter of the standard card.

Ps = 2l + 2w

Ps = 2(3.5) + 2(2.5)

Ps = 12 in.

Find the scale factor.

Pm = Ps · k

6 = 12 · k

1 _ 2

= k

Find the area of the standard card.

A s = l s · w s

A s = 3.5 · 2.5

A s = 8.75 in 2

Find the area of the mini-card.

A m = A s · k 2

A m = 8.75 · ( 1 _ 2

) 2

A m = 8.75 · 1 _ 4

A m = 2.1875 in 2

The area of the mini card is about 2.2 square inches.

Justify and Evaluate

To find the area of the mini-card, find its length and width by multiplying

the dimensions of the standard card by the scale factor. The length of the

mini-card is l s · 1 _ 2 = 3.5 · 1 _

2 = 1.75 in., and the width is w s · 1 _

2 = 2.5 · 1 _

2 =

1.25 in. So, A m = l m · w m = 1.75 · 1.25 = 2.1875 in 2 . The answer is correct.

STEP 1

STEP 2

STEP 3

STEP 4

Use the formula for the area of a rectangle.

Multiply the perimeter of the standard card by the scale factor to get the perimeter of the mini-card.

Multiply the area of the standard card by the scale factor squared to get the area of the mini-card.

377Lesson 13.3

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Math Trainer

Online Assessment and Intervention

Personal

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Find the perimeter and area of the image after dilating the figures shown

with the given scale factor. (Explore Activity and Example 1)

1. Scale factor = 5

P = 12 A = 9

P' = A' =

2. Scale factor = 3 _ 4

P = 48 A = 128

P' = A' =

A group of friends is roping off a soccer field in a back yard. A full-size

soccer field is a rectangle with a length of 100 yards and a width of 60 yards.

To fit the field in the back yard, the group needs to reduce the size of the

field so its perimeter is 128 yards. (Example 1)

3. What is the perimeter of the full-size soccer field?

4. What is the scale factor of the dilation?

5. What is the area of the soccer field in the back yard?

6. When a rectangle is dilated, how do the perimeter and area

of the rectangle change?

ESSENTIAL QUESTION CHECK-IN??

Guided Practice

2. Johnson Middle School is selling mouse pads that are replicas of a student’s

award-winning artwork. The rectangular mouse pads are dilated from the

original artwork and have a length of 9 inches and a width of 8 inches.

The perimeter of the original artwork is 136 inches. What is the area of the

original artwork?

YOUR TURN

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Name Class Date

Independent Practice13.3

7. When you make a photocopy of an image, is the photocopy a dilation?

What is the scale factor? How do the perimeter and area change?

8. Problem Solving The universally accepted film size for movies has a

width of 35 millimeters. If you want to project a movie onto a square

sheet that has an area of 100 square meters, what is the scale factor that is

needed for the projection of the movie? Explain.

9. The perimeter of a square is 48 centimeters. If the square is dilated by a

scale factor of 0.75, what is the length of each side of the new square?

10. The screen of an eReader has a length of 8 inches and a width of 6 inches.

Can the page content from an atlas that measures 19 inches by 12 inches

be replicated in the eReader? If not, propose a solution to move the atlas

content into the eReader format.

11. Represent Real-World Problems There are 64 squares on a chessboard.

Each square on a tournament chessboard measures 2.25 × 2.25 inches. A

travel chessboard is a dilated replica of the tournament chessboard using a

scale factor of 1 _ 3 .

a. What is the size of each square on the travel chessboard?

b. How long is each side of the travel board?

c. How much table space do you need to play on the travel chessboard?

8.3.B, 8.10.A, 8.10.B, 8.10.D

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Work Area

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W X

12. Draw Conclusions The legs of a right triangle are 3 units and 4 units

long. Another right triangle is dilated from this triangle using a scale factor

of 3. What are the side lengths and the perimeter of the dilated triangle?

13. Critique Reasoning Rectangle W′X′Y′Z′ below is a dilation of rectangle

WXYZ. A student calculated the area of rectangle W'X'Y'Z' to be 36 square

units. Do you agree with this student's calculation? If not, explain and

correct the mistake.

14. Multistep Rectangle A'B'C'D' is a dilation of rectangle ABCD, and the scale

factor is 2. The perimeter of ABCD is 18 mm. The area of ABCD is 20 mm2.

a. Write an equation for, and calculate, the perimeter of A'B'C'D'.

b. Write an equation for, and calculate, the area of A'B'C'D'.

c. The side lengths of both rectangles are whole numbers of millimeters.

What are the side lengths of ABCD and A'B'C'D'?

FOCUS ON HIGHER ORDER THINKING

Unit 5380

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