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

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

3. GUIDED PRACTICE for Example 1 SOLUTION J = ~ ~~ == P,P, K Q LR and The congruent angles are JK PQ = KL QR = LJ RP The ratios of the corresponding side lengths are. 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.

4. 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.

5. EXAMPLE 2 Find the scale factor SOLUTION STEP 1 Identify pairs of congruent angles. From the diagram, you can see that Z F, Y G, and X H. Angles W and J are right angels, so W J. So, the corresponding angles are congruent.

6. EXAMPLE 2 Find the scale factor SOLUTION STEP 2 Show that corresponding side lengths are proportional. XW HJ ZY FG YX GH WZ JF 25 20 = 15 12 = 5 4 = 20 16 == 5 4 5 4 = = 5 4 30 24 =

7. EXAMPLE 2 Find the scale factor SOLUTION The ratios are equal, so the corresponding side lengths are proportional. So ZYXW ~ FGHJ. The scale factor of ZYXW to FGHJ is ANSWER 5 4.

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

9. EXAMPLE 3 Use similar polygons Write proportion. Substitute. Cross Products Property Solve for x. SOLUTION The triangles are similar, so the corresponding side lengths are proportional. x = 15 12x = 180 MN DE NP EF = = 12 9 20 x

10. GUIDED PRACTICE for Examples 2 and 3 In the diagram, ABCD ~ QRST. SOLUTION STEP 1 Identify pairs of congruent angles. From the diagram, you can see that A = Q, T = D, and B = R. Angles C and S are right angles. So, all the corresponding angles are congruent. 2. What is the scale factor of QRST to ABCD ?

11. GUIDED PRACTICE for Examples 2 and 3 STEP 2 Show that corresponding side lengths are proportional. QR AB QT AD TS DC RS BC 5 10 = 6 12 = 1 2 = 8 16 = 4 x == 1 2 1 2 =

12. GUIDED PRACTICE for Examples 2 and 3 The ratios are equal, so the corresponding side lengths are proportional. ANSWER So QRST ~ ABCD. The scale factor of QRST to ABCD 1 2.

13. GUIDED PRACTICE for Examples 2 and 3 3. Find the value of x. In the diagram, ABCD ~ QRST. Write proportion. Substitute. Cross Products Property Solve for x. SOLUTION The triangles are similar, so the corresponding side lengths are proportional. RS QS BC AC = 4 6 4 + = x 12 x + 4(12 + x) =10 x x = 8

14. GUIDED PRACTICE for Examples 2 and 3 ANSWER So the value of x is 8

15. 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.

16. EXAMPLE 4 Find perimeters of similar figures SOLUTION Because the new pool will be similar to an Olympic pool, the scale factor is the ratio of the lengths, a. 40 50 = 4 5 Find the perimeter of an Olympic pool and the new pool. b.

17. EXAMPLE 4 Find perimeters of similar figures x 150 4 5 = Use Theorem 6.1 to write a proportion. x = 120 Multiply each side by 150 and simplify. The perimeter of the new pool is 120 meters. ANSWER The perimeter of an Olympic pool is 2(50) + 2(25) = 150 meters. You can use Theorem 6.1 to find the perimeter x of the new pool. b.

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

19. GUIDED PRACTICE for Example 4 5. Find the value of x. In the diagram, ABCDE ~ FGHJK. x 18 10 15 = Use Theorem 6.1 to write a proportion. Cross product property. You can use the theorem 6.1 to find the perimeter of x x = 12 SOLUTION 15 x = 18 10

20. GUIDED PRACTICE for Example 4 ANSWER The value of x is 12

21. GUIDED PRACTICE for Example 4 6. Find the perimeter of ABCDE. In the diagram, ABCDE ~ FGHJK. SOLUTION As the two polygons are similar the corresponding side lengths are similar To find the perimeter of ABCDE first find its’ side lengths.

22. GUIDED PRACTICE for Example 4 FG AB = FK AE Write Equation Substitute 15x = 180 Cross Products Property x = 12 Solve for x AE = 12 To find AE 15 10 = 18 x

23. GUIDED PRACTICE for Example 4 Write Equation 15 10 = 15 y Substitute 15y = 150 Cross Products Property y = 10 Solve for y ED = 10 To find ED FG AB = KJ ED

24. GUIDED PRACTICE for Example 4 FG AB = HJ CD Write Equation 15 10 = 12 z Substitute 15z = 120 Cross Products Property z = 8 Solve for z DC = 8 To find DC

25. GUIDED PRACTICE for Example 4 FG AB = GH BC Write Equation 15 10 = 9 a Substitute 15a = 90 Cross Products Property a = 6 Solve for x BC = 6 To find BC

26. GUIDED PRACTICE for Example 4 The perimeter of ABCDE = AB + BC + CD + DE + EA = 10 + 6 + 8 + 10 + 12 = 46 ANSWER The perimeter of ABCDE = 46

27. EXAMPLE 5 Use a scale factor In the diagram, ∆ TPR ~ ∆ XPZ. Find the length of the altitude PS. SOLUTION First, find the scale factor of ∆ TPR to ∆ XPZ. TR XZ 6 + 6 = 8 + 8 = 12 16 = 3 4

28. EXAMPLE 5 Use a scale factor Because the ratio of the lengths of the altitudes in similar triangles is equal to the scale factor, you can write the following proportion. The length of the altitude PS is 15. Write proportion. Substitute 20 for PY. Multiply each side by 20 and simplify. PS PY 3 4 = PS 20 3 4 = = PS 15 ANSWER

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

30. GUIDED PRACTICE for Example 5 JL EG First find the scale factor of ∆ JKL to ∆ EFG. SOLUTION 48 + 48 = 40 + 40 = 96 80 = 6 5 Because the ratio of the lengths of the median in similar triangles is equal to the scale factor, you can write the following proportion.

31. GUIDED PRACTICE for Example 5 SOLUTION KM = 42 KM HF = 6 5 KM 35 = 6 5 Write proportion. Substitute 35 for HF. Multiply each side by 35 and simplify.