Perimeters and Areas of Similar Figures

Geometry

Perimeters and Areas of Similar Figures

May 8, 2023 2

Screen Size: 6 9Screen Size: 12 18

The picture is twice as big as our smaller TV.

Area: 54Area: 216

Is this Is this statement statement True or True or False?False?

May 8, 2023 3

Goals

Compare perimeters and areas of similar figures.

Use perimeters and areas of similar figures to solve problems.

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Problem

Draw a rectangle that measures 3 2.

The area is ?

6

Double the lengths of the sides and draw a rectangle that measures 6 4.The area is ? 24

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Example continued

6When the lengths of the sides were doubled, did the area double?

NO!24

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So what happened?

24

136 2

6

24

23

4

6

The ratio of the sides:

The ratio of the areas:6

2414

?

?

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Notice:

1214

6

24

23

4

6

The ratio of the sides:

The ratio of the areas:

21 12 4

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Theorem 10.5

The Area of Similar Polygons If two polygons are similar with

lengths of corresponding sides in the ratio a:b, then the ratio of their areas is a2:b2.

Another way of saying it: the ratio of the areas is the square of the ratio of the sides.

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Example 1 ABCDE ~ MNOPQ

A

B

C D

E

M

N

O P

Q46

Find the ratio of the areas of the figures.

2 24 26 3

49

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Example 2

410

25

4

10

These figures are similar. The area of the smaller one is 20. Find the area of the larger figure.Ratio of the sides:

Ratio of their areas: 22 4

25 5

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Example 2 continued

4

10

Write the proportion and solve:

20 425

4 20(25)4 500

125

xxxx

20 x

Ratio of sides 2:5Ratio of areas 4:25

125

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Your Turn

5

8

These figures are similar. The area of the larger one is 320. Find the area of the smaller figure.

320

?

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Solution

5

8320

?

258 3202564 320

64 25(320)64 8000

800064

125

x

x

xx

x

x

125

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Now do you see how it works?

5 1258 320Do NOT write

5(320) = 1600 and 8(125)=1000

Instead write 25 12564 320

25(320) = 8000 and 64(125) = 8000

Ratio of sides.

Ratio of SQUARE of sides.

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Try it again…These figures are similar. Find the area of the smaller figure if the area of the larger one is 720.5

12A = 720

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Solution

5

12A = 720

2512 72025144 720144 25(720)144 18000

125

x

x

xxx

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Problem

A = 36

A = 100

4

x

If the figures are similar, what is the length of the side marked x?

Remember: the ratio of the areas is the square of the ratios of the sides.Or, the ratio of the sides is the square root of the ratios of the areas.

May 8, 2023 18

Problem Solution

A = 36

A = 100

4

x

The ratio of the sides is the square root of the ratios of the areas.

2

23

4 36100

4 610

6 40406

6

x

xx

x

x

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Here’s nothing surprising:

If two polygons are similar, the ratio of the perimeters is equal to the ratio of the sides.

23

4

6

Perimeter = 10 Perimeter = 2010 120 2

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Example

ABCD ~ DEFG. AB = 8 and DE = 12. The perimeter of ABCD is 20. What is the perimeter of DEFG? Solution: 8 20

128 12(20)8 240

30

xxxx

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Summary

The ratio of the areas of similar figures is the square of the ratio of the sides.

The ratio of the perimeters of similar figures is the same as the ratio of the sides.

The ratio of the sides in two similar figures is the square root of the ratio of the areas.

May 8, 2023 22

Do you get it? Do these problems.

1. Figure CAT is similar to BAT.CA = 10 and BA = 15.

a. What is the ratio of the sides?

b. What is the ratio of the perimeters of CAT to BAT?

c. What is the ratio of the areas of CAT to BAT?

2/3

2/3

4/9

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2. Figure RATS is similar to DOGS.RA = 12 and DO = 30.

a. What is the ratio of the perimeters of RATS to DOGS?

b. What is the ratio of the areas of RATS to DOGS?

25425

May 8, 2023 24

3. Two figures, ABCDE and XYZWS are similar. The perimeter of ABCDE is and the perimeter of XYZWS is 18. What is the ratio of corresponding sides? (What is the scale factor?)

3 3

3 318

36

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4. JKL ~ RST. The area of JKL is 25 and the area of RST is 64. What is the ratio of their perimeters?

2564

58

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5. HOT ~ DOG. The area of HOT is 100 and the area of DOG is 9. The perimeter of HOT is 40. What is the perimeter of DOG?Ratio of Areas: Ratio of Perimeters:

1009

103

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4. HOT ~ DOG. The area of HOT is 100 and the area of DOG is 9. The perimeter of HOT is 40. What is the perimeter of DOG?

Ratio of Perimeters:103

10 40310 120

12

xxx