Course 3

5-5 Similar Figures5-5 Similar Figures

Course 3

Warm UpWarm Up

Problem of the DayProblem of the Day

Lesson PresentationLesson Presentation

Course 3

5-5 Similar Figures

Warm UpSolve each proportion.

b30

39

=1. 5635

y5

=2.

412

p9

=3. 56m

2826

=4.

b = 10 y = 8

p = 3 m = 52

Course 3

5-5 Similar Figures

Problem of the Day

A rectangle that is 10 in. wide and 8 in. long is the same shape as one that is 8 in. wide and x in. long. What is the length of the smaller rectangle?6.4 in.

Course 3

5-5 Similar Figures

Learn to determine whether figures are similar, to use scale factors, and to find missing dimensions in similar figures.

Course 3

5-5 Similar Figures

Vocabulary

similar

congruent angles

scale factor

Course 3

5-5 Similar Figures

Similar figures have the same shape, but not necessarily the same size. Two triangles are similar if the ratios of the lengths of corresponding sides are equivalent and the corresponding angles have equal measures. Angles that have equal measures are called congruent angles.

43°

43°

43°

Course 3

5-5 Similar Figures

Additional Example 1: Identifying Similar Figures

Which rectangles are similar?

Since the three figures are all rectangles, all the angles are right angles. So the corresponding angles are congruent.

Course 3

5-5 Similar Figures

Additional Example 1 Continued

50 48

The ratios are equal. Rectangle J is similar to rectangle K. The notation J ~ K shows similarity.

The ratios are not equal. Rectangle J is not similar to rectangle L. Therefore, rectangle K is not similar to rectangle L.

20 = 20

length of rectangle Jlength of rectangle K

width of rectangle Jwidth of rectangle K

10 5

42

? =

length of rectangle Jlength of rectangle L

width of rectangle Jwidth of rectangle L

1012

45

?=

Compare the ratios of corresponding sides to see if they are equal.

Course 3

5-5 Similar Figures

Check It Out: Example 1

Which rectangles are similar?

A8 ft

4 ft

B6 ft

3 ft

C5 ft

2 ft

Since the three figures are all rectangles, all the angles are right angles. So the corresponding angles are congruent.

Course 3

5-5 Similar Figures

Check It Out: Example 1 Continued

16 20

The ratios are equal. Rectangle A is similar to rectangle B. The notation A ~ B shows similarity.

The ratios are not equal. Rectangle A is not similar to rectangle C. Therefore, rectangle B is not similar to rectangle C.

24 = 24

length of rectangle Alength of rectangle B

width of rectangle Awidth of rectangle B

8 6

43

? =

length of rectangle Alength of rectangle C

width of rectangle Awidth of rectangle C

8 5

42

?=

Compare the ratios of corresponding sides to see if they are equal.

Course 3

5-5 Similar Figures

The ratio formed by the corresponding sides is the scale factor.

Course 3

5-5 Similar Figures

Additional Example 2A: Using Scale Factors to Find Missing Dimensions

A picture 10 in. tall and 14 in. wide is to be scaled to 1.5 in. tall to be displayed on a Web page. How wide should the picture be on the Web page for the two pictures to be similar?

Divide the height of the scaled picture by the corresponding height of the original picture.

0.151.510

=

Multiply the width of the original picture by the scale factor.

2.1

14 • 0.15

The picture should be 2.1 in. wide.

Simplify.

Course 3

5-5 Similar Figures

Additional Example 2B: Using Scale Factors to Find Missing Dimensions

A toy replica of a tree is 9 inches tall. What is the length of the tree branch, if a 6 ft tall tree has a branch that is 60 in. long?

Divide the height of the replica tree by the height of the 6 ft tree in inches, to find the scale factor.

.125972

=

Multiply the length of the original tree branch by the scale factor.

7.5

60 • .125

The toy replica tree branch should be 7.5 in long.

Simplify.

Course 3

5-5 Similar Figures

Check It Out: Example 2A

Divide the height of the scaled picture by the corresponding height of the original picture.

0.251040

=

Multiply the width of the original picture by the scale factor.

14

56 • 0.25

The picture should be 14 in. wide.

Simplify.

A painting 40 in. tall and 56 in. wide is to be scaled to 10 in. tall to be displayed on a poster. How wide should the painting be on the poster for the two pictures to be similar?

Course 3

5-5 Similar Figures

Check It Out: Example 2B

A toy replica of a tree is 6 inches tall. What is the length of the tree branch, if a 8 ft tall tree has a branch that is 72 in. long?

Divide the height of the replica tree by the height of the 6 ft tree in inches, to find the scale factor.

.0625696

=

Multiply the length of the original tree branch by the scale factor.

4.5

72 • .0625

The toy replica tree branch should be 4.5 in long.

Simplify.

Course 3

5-5 Similar Figures

Additional Example 3: Using Equivalent Ratios to Find Missing Dimensions

A T-shirt design includes an isosceles triangle with side lengths 4.5 in, 4.5 in., and 6 in. An advertisement shows an enlarged version of the triangle with two sides that are each 3 ft. long. What is the length of the third side of the triangle in the advertisement?

Set up a proportion.6 in.x ft

4.5 in.3 ft

=

4.5 in. • x ft = 3 ft • 6 in. Find the cross products.

4.5 in. • x ft = 3 ft • 6 in. Divide out the units.

Course 3

5-5 Similar Figures

Draw a diagram to help you visualize the problems

Helpful Hint

Course 3

5-5 Similar Figures

4.5x = 3 • 6

4.5x = 18

x = = 4184.5

Cancel the units.

Multiply.

Solve for x.

Additional Example 3 Continued

The third side of the triangle is 4 ft long.

Course 3

5-5 Similar Figures

Check It Out: Example 3

Set up a proportion.24 ftx in.

18 ft4 in.

=

18 ft • x in. = 24 ft • 4 in. Find the cross products.

18 ft • x in. = 24 ft • 4 in. Divide out the units.

A flag in the shape of an isosceles triangle with side lengths 18 ft, 18 ft, and 24 ft is hanging on a pole outside a campground. A camp t-shirt shows a smaller version of the triangle with two sides that are each 4 in. long. What is the length of the third side of the triangle on the t-shirt?

Course 3

5-5 Similar Figures

18x = 24 • 4

18x = 96

x = 5.39618

Cancel the units.

Multiply.

Solve for x.

Check It Out: Example 3 Continued

The third side of the triangle is about 5.3 in. long.

Course 3

5-5 Similar Figures

Lesson QuizUse the properties of similar figures to answer each question.

1. A rectangular house is 32 ft wide and 68 ft long. On a blueprint, the width is 8 in. Find the length on the blueprint.

2. Karen enlarged a 3 in. wide by 5 in. tall photo into a poster. If the poster is 2.25 ft wide, how tall is it?

3. Which rectangles are similar?

17 in.

3.75 ft

A and B are similar.