5-5 Similar Figures

Preparation for MG1.2 Construct and read drawings and models made to scale.

California Standards

5-5 Similar Figures

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

Corresponding sides of two figures are in the same relative position, and corresponding angles are in the same relative position. Two figures are similar if the lengths of corresponding sides are proportional and the corresponding angles have equal measures.

5-5 Similar Figures

43°

43°

5-5 Similar Figures

A is read as “angle A.” ∆ABC is read as “triangle ABC.” “∆ABC ~∆EFG” is read as “triangle ABC is similar to triangle EFG.”

Reading Math

5-5 Similar FiguresAdditional 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.

5-5 Similar FiguresAdditional 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.

5-5 Similar FiguresCheck 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.

5-5 Similar FiguresCheck 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.

5-5 Similar Figures

14 ∙ 1.5 = w ∙ 10 Find the cross products.

Divide both sides by 10.10w10

2110

=

width of a picturewidth on Web page

height of picture height on Web page

14w

= 101.5

2.1 = w

Set up a proportion. Let w be the width of the picture on the Web page.

The picture on the Web page should be 2.1 in. wide.

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?

Additional Example 2: Finding Missing Measures in Similar Figures

21 = 10w

5-5 Similar Figures

56 ∙ 10 = w ∙ 40 Find the cross products.

Divide both sides by 40.40w40

56040

=

width of a paintingwidth of poster

length of painting length of poster

56w

= 4010

14 = w

Set up a proportion. Let w be the width of the painting on the Poster.

The painting displayed on the poster should be 14 in. long.

Check It Out! Example 2

560 = 40w

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

5-5 Similar FiguresAdditional Example 3: Business Application

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.3 ftx ft

4.5 in.6 in.

=

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

side of small triangle base of small triangle

side of large triangle base of large triangle

=

4.5x = 18 Multiply.

5-5 Similar Figures

x = = 4184.5

Solve for x.

Additional Example 3 Continued

The third side of the triangle is 4 ft long.

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?

5-5 Similar FiguresCheck 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.

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?side of large triangle side of small triangle

base of large triangle base of small triangle

=

18x = 96 Multiply.

5-5 Similar Figures

x = 5.39618

Solve for x.

Check It Out! Example 3 Continued

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

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?