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7-4 Similar Figures7-4 Similar Figures

Course 1

Warm UpWarm Up

Lesson PresentationLesson Presentation

Problem of the DayProblem of the Day

Course 1

7-4 Similar Figures

Warm UpFill in the missing value.

1. c = 2 qt

2. 180 in. = yd

3. 3 tons = lb

4. min = 2,760 s

8

5

6,000

46

Course 1

7-4 Similar Figures

Problem of the Day

How many 8 in. by 10 in. rectangular tiles would be needed to cover a 16 ft by 20 ft floor?

576

Course 1

7-4 Similar Figures

Learn to use proportions to find missing measures in similar figures.

Course 1

7-4 Similar Figures

Vocabulary

corresponding sidescorresponding anglessimilar

Course 1

7-4 Similar Figures

Matching sides of two or more polygons are called corresponding sides, and matching angles are called corresponding angles.

Course 1

7-4 Similar Figures

Similar Figures

Two figures are similar if• the measures of the corresponding angles are equal• the ratios of the lengths of the corresponding sides t are proportional

Similar figures have the same shape but not necessarily the same size.

Course 1

7-4 Similar Figures

Additional Example 1: Finding Missing Measures in Similar Figures

111 y

___ 100 200

____ = Write a proportion using corresponding side lengths.

The cross products are equal.200 • 111 = 100 • y

The two triangles are similar. Find the missing length y and the measure of D.

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7-4 Similar Figures

Additional Example 1 Continued

y is multiplied by 100.22,200 = 100y

22,200 100

______ 100y 100

____ = Divide both sides by 100 to undo the multiplication.

222 mm = y

Angle D is congruent to angle C, and m C = 70°.

m D = 70°

The two triangles are similar. Find the missing length y and the measure of D.

Course 1

7-4 Similar Figures

Check It Out: Example 1

52 y

___ 50 100

____ = Write a proportion using corresponding side lengths.

The cross products are equal.100 • 52 = 50 • y

A B60 m 120 m

50 m 100 m

y52 m

65°

45°

The two triangles are similar. Find the missing length y and the measure of B.

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7-4 Similar Figures

Check It Out: Example 1 Continued

y is multiplied by 50.5,200 = 50y

5,200 50

_____ 50y 50

___ = Divide both sides by 50 to undo the multiplication.

104 m = y

Angle B is congruent to angle A, and m A = 65°.

m B = 65°

The two triangles are similar. Find the missing length y and the measure of B.

Course 1

7-4 Similar Figures

This reduction is similar to a picture that Katie painted. The height of the actual painting is 54 centimeters. What is the width of the actual painting?

11 Understand the ProblemThe answer will be the width of the actual painting.

List the important information:

• The actual painting and the reduction above are t similar.

• The reduced painting is 2 cm tall and 3 cm wide.

• The actual painting is 54 cm tall.

Additional Example 2: Problem Solving Application

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7-4 Similar Figures

Additional Example 2 Continued

Draw a diagram to represent the situation. Use the corresponding sides to write a proportion.

22 Make a Plan

ReducedActual

2 54

3w

Course 1

7-4 Similar Figures

Additional Example 2 Continued

Solve33

54 • 3 = 2 • w

162 = 2w

162 2

____ 2w 2

___ =

81 = w

The width of the actual painting is 81 cm.

Write a proportion.

The cross products are equal.

w is multiplied by 2.

Divide both sides by 2 to undo the multiplication.

3 cm w cm

2 cm 54 cm

_____ =

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7-4 Similar Figures

Additional Example 2 Continued

Look Back44

Estimate to check your answer. The ratio of the heights is about 2:50 or 1:25. The ratio of the widths is about 3:90, or 1:30. Since these ratios are close to each other, 81 cm is a reasonable answer.

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7-4 Similar Figures

Check It Out: Example 2This reduction is similar to a picture that Marty designed. The height of the actual picture is 39 inches. What is the width of the actual picture?

11 Understand the ProblemThe answer will be the width of the actual painting.

List the important information:

• The actual painting and the reduction above are t similar.

• The reduced painting is 3 in. tall and 4 in. wide.

• The actual painting is 39 in. tall.

4 in.

3 in.

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7-4 Similar Figures

Check It Out: Example 2 Continued

Draw a diagram to represent the situation. Use the corresponding sides to write a proportion.

22 Make a Plan

ReducedActual

3 39

4w

Course 1

7-4 Similar Figures

Check It Out: Example 2 Continued

Solve33 4 in w in

____ 3 in 39 in

_____ =

39 • 4 = 3 • w

156 = 3w

156 3

____ 3w 3

___ =

52 = w

The width of the actual painting is 52 inches.

Write a proportion.

The cross products are equal.

w is multiplied by 3.

Divide both sides by 3 to undo the multiplication.

Course 1

7-4 Similar Figures

Check It Out: Example 2 Continued

Look Back44

Estimate to check your answer. The ratio of the heights is about 4:40, or 1:10. The ratio of the widths is about 5:50, or 1:10. Since these ratios are the same, 52 inches is a reasonable answer.

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7-4 Similar Figures

Lesson QuizThese two triangles are similar.

1. Find the missing length x.

2. Find the measure of J.

3. Find the missing length y.

4. Find the measure of P.

5. Susan is making a wood deck from plans for an

8 ft by 10 ft deck. However, she is going to

increase its size proportionally. If the length is

to be 15 ft, what will the width be?

36.9°

30 in.

4 in.

90°

12 ft