Use proportions to identify similar polygons. 6.3 Use Similar Polygons Use proportions to identify similar polygons.

Vocabulary Two polygons are similar polygons if corresponding angles are congruent and corresponding side lengths are proportional.

EXAMPLE 1 Use similarity statements In the diagram, ∆RST ~ ∆XYZ a. List all pairs of congruent angles. b. Check that the ratios of corresponding side lengths are equal. c. Write the ratios of the corresponding side lengths in a statement of proportionality.

EXAMPLE 1 Use similarity statements SOLUTION a. R = ~ X, S Y T Z and RS XY = 20 12 5 3 b. ; ST = 30 18 5 3 YZ ; TR ZX = 25 15 5 3 c. Because the ratios in part (b) are equal, YZ RS XY = ST TR ZX .

The congruent angles are GUIDED PRACTICE for Example 1 L J K R P Q 1. Given ∆ JKL ~ ∆ PQR, list all pairs of congruent angles. Write the ratios of the corresponding side lengths in a statement of proportionality. SOLUTION J = ~ P, K Q L R and The congruent angles are JK PQ = KL QR LJ RP The ratios of the corresponding side lengths are.

Vocabulary If two polygons are similar, then the ratio of the lengths or two corresponding sides is the called the scale factor.

EXAMPLE 2 Find the scale factor Determine whether the polygons are similar. If they are, write a similarity statement and find the scale factor of ZYXW to FGHJ.

EXAMPLE 3 Use similar polygons In the diagram, ∆DEF ~ ∆MNP. Find the value of x. ALGEBRA x = 15

GUIDED PRACTICE for Examples 2 and 3 In the diagram, ABCD ~ QRST. 2. What is the scale factor of QRST to ABCD ?

GUIDED PRACTICE for Examples 2 and 3 In the diagram, ABCD ~ QRST. 3. Find the value of x. x = 8

Theorem 6.1: Perimeters of Similar Polygons If two polygons are similar, then the ratio of their perimeters is equal to the ratios of their corresponding side lengths.

EXAMPLE 4 Find perimeters of similar figures Swimming A town is building a new swimming pool. An Olympic pool is rectangular with length 50 meters and width 25 meters. The new pool will be similar in shape, but only 40 meters long. Find the scale factor of the new pool to an Olympic pool. a. Find the perimeter of an Olympic pool and the new pool. b.

GUIDED PRACTICE for Example 4 In the diagram, ABCDE ~ FGHJK. 4. Find the scale factor of FGHJK to ABCDE. The scale factor is the ratio of the length is ANSWER 15 10 = 3 2

GUIDED PRACTICE for Example 4 In the diagram, ABCDE ~ FGHJK. 5. Find the value of x. x = 12

The perimeter of ABCDE is 46 units GUIDED PRACTICE for Example 4 In the diagram, ABCDE ~ FGHJK. 6. Find the perimeter of ABCDE. The perimeter of ABCDE is 46 units

Similarity and Congruence Notice that any two congruent figures are also similar. Their scale factor is 1:1. In triangles ABC and DEF, the scale factor is 5:5 = 1.

Corresponding Lengths You know that perimeters of similar polygons are in the same ratio as a corresponding side lengths. You can extend this concept to other segments in polygons.

Key Concept: Corresponding Lengths in Similar Polygons If two polygons are similar, then the ratio of any two corresponding lengths in the polygons is equal to the scale factor of the similar polygons.

EXAMPLE 5 Use a scale factor In the diagram, ∆TPR ~ ∆XPZ. Find the length of the altitude PS . ANSWER The length of the altitude PS is 15.

GUIDED PRACTICE for Example 5 In the diagram, ABCDE ~ FGHJK. In the diagram, ∆JKL ~ ∆ EFG. Find the length of the median KM. 7. KM = 42

 Homework  Page 376/1,2,4-10,14-18,23-26,29-32