4-8 Similar Figures and Proportions

Warm UpWarm Up

Lesson PresentationLesson Presentation

Problem of the DayProblem of the Day

Lesson QuizzesLesson Quizzes

4-8 Similar Figures and Proportions

Warm UpFind the cross products, and then tell whether the ratios are equal.

1. 166

, 4015

2. 38

, 1846

3. 89

, 2427

4. 2812

, 4218

240 = 240; equal

216 = 216; equal

504 = 504; equal

138 = 144; not equal

4-8 Similar Figures and Proportions

Problem of the DayEvery 8th telephone pole along a road has a red band painted on it. Every 14th pole has an emergency call phone on it. What is the number of the first pole with both a red band and a call phone?

56

4-8 Similar Figures and Proportions

Learn to use ratios to determine if two figures are similar.

4-8 Similar Figures and Proportions

Vocabulary

similar

corresponding sides

corresponding angles

4-8 Similar Figures and Proportions

Similar figures are figures that have the same shape but not necessarily the same size. The symbol ~ means “is similar to.”

4-8 Similar Figures and Proportions

Corresponding angles of two or more similar polygons are in the same relative position. Corresponding sides of two or more similar polygons are in the same relative position. When naming similar figures, list the corresponding angles in the same order. For the triangles below, ∆ABC ~ ∆DEF.

D

E

FA

B

C

4-8 Similar Figures and Proportions

Two figures are similar if• The measures of their corresponding angles are

equal. • The ratios of the lengths of the corresponding

sides are proportional.

SIMILAR FIGURES

4-8 Similar Figures and Proportions

A side of a figure can be named by its endpoints, with a bar above such as;

AB

Without the bar, the letters indicate the length of the side.

Reading Math

4-8 Similar Figures and Proportions

Identify the corresponding sides in the pair of triangles. Then use ratios to determine whether the triangles are similar.

Additional Example 1: Determining Whether Two Triangles Are Similar

A C

B

10 in

4 in 7 in D

E

F

16 in 28 in

40 in

AB corresponds to DE.

BC corresponds to EF.

ABDE

=? BCEF

=? ACDF

416

728

1040

14

14

14

Since the ratios of the corresponding sides are equivalent, the triangles are similar.

Write ratios using the corresponding sides.

Substitute the length of the sides.

Simplify each ratio.

=? =?

AC corresponds to DF.

=? =?

4-8 Similar Figures and Proportions

Identify the corresponding sides in the pair of triangles. Then use ratios to determine whether the triangles are similar.

Check It Out: Example 1

A C

B

9 in

3 in 7 in D

E

F

9 in 21 in

27 in

AB corresponds to DE.

BC corresponds to EF.

ABDE

=? BCEF

=? ACDF

39

721

927

13

13

13

Since the ratios of the corresponding sides are equivalent, the triangles are similar.

Write ratios using the corresponding sides.

Substitute the length of the sides.

Simplify each ratio.

=? =?

AC corresponds to DF.

=? =?

4-8 Similar Figures and Proportions

Tell whether the figures are similar.

Additional Example 2: Determining Whether Two Four-Sided Figures are Similar

The corresponding angles of the figures have equal measure.

Write each set of corresponding sides as a ratio.

4-8 Similar Figures and Proportions

Additional Example 2 Continued

MNQR MN corresponds to QR.

NORS

OPST

MPQT

NO corresponds to RS.

OP corresponds to ST.

MP corresponds to QT.

4-8 Similar Figures and Proportions

Determine whether the ratios of the lengths of the corresponding sides are proportional.

Additional Example 2 Continued

Write ratios using corresponding sides.

Substitute the length of the sides.

Simplify each ratio.

Since the ratios of the corresponding sides are equivalent, the figures are similar.

MNQR

=? NORS

=? OPST

=? MPQT

6 9=

? 812 =

? 4 6 =

? 1015

23

= 23

= 23

= 23

? ? ?

4-8 Similar Figures and Proportions

Tell whether the figures are similar.

Check It Out: Example 2

The corresponding angles of the figures have equal measure.

100 m

80 m

60 m 47.5 m80°

90°125°

65°M P

NO

400 m

320 m

190 m240 m

80° 65°

90°125°

Q T

R S

Write each set of corresponding sides as a ratio.

4-8 Similar Figures and Proportions

Check It Out: Example 2 Continued

MNQR MN corresponds to QR.

NORS

OPST

MPQT

NO corresponds to RS.

OP corresponds to ST.

MP corresponds to QT.

100 m

80 m

60 m 47.5 m80°

90°125°

65°M P

NO

400 m

320 m

190 m240 m

80° 65°

90°125°

Q T

R S

4-8 Similar Figures and Proportions

Determine whether the ratios of the lengths of the corresponding sides are proportional.

Check It Out: Example 2 Continued

Write ratios using correspondingsides.

Substitute the length of the sides.

Simplify each ratio.

Since the ratios of the corresponding sides are equivalent, the figures are similar.

100 m

80 m

60 m 47.5 m80°

90°125°

65°M P

NO

400 m

320 m

190 m240 m

80° 65°

90°125°

Q T

R S

MNQR

=? NORS

=? OPST

=? MPQT

60240 =

? 80320 =

? 47.5190 =

? 100400

14

= 14

= 14

= 14

? ? ?

4-8 Similar Figures and Proportions

Standard Lesson Quiz

Lesson Quizzes

Lesson Quiz for Student Response Systems

4-8 Similar Figures and Proportions

Lesson Quiz: Part I

Tell whether the figures are similar.

1.similar

59°59°

35°35°

86°86°

4-8 Similar Figures and Proportions

Lesson Quiz: Part II

Tell whether the figures are similar.

2.

not similar

119°

55°

107°

79°

107°

80°

135°

38°

4-8 Similar Figures and Proportions

1. Which of the following triangles are similar?

A. C.

B. D.

Lesson Quiz for Student Response Systems

4-8 Similar Figures and Proportions

2. Which of the following figures are similar?

A.

B.

Lesson Quiz for Student Response Systems