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EXAMPLE 1

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. a. 5. ST.

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EXAMPLE 1

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  1. 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

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

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

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

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

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

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

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

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

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

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

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

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

  14. ANSWER So the value of x is8 for Examples 2 and 3 GUIDED PRACTICE

  15. 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. a. Find the scale factor of the new pool to an Olympic pool. EXAMPLE 4 Find perimeters of similar figures

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

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

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

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

  20. ANSWER The value ofxis12 for Example 4 GUIDED PRACTICE

  21. In the diagram, ABCDE ~ FGHJK. for Example 4 GUIDED PRACTICE 6. Find the perimeter of ABCDE. 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. FG FK AB AE 15 18 = = 10 x for Example 4 GUIDED PRACTICE To find AE Write Equation Substitute 15x = 180 Cross Products Property x = 12 Solve for x AE = 12

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

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

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

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

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

  28. = 3 PS 3 4 4 PY PS = 20 = PS 15 The length of the altitude PSis 15. 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. Write proportion. Substitute 20 for PY. Multiply each side by 20 and simplify. ANSWER

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

  30. 48 + 48 = = = 40 + 40 JL 96 6 EG 80 5 for Example 5 GUIDED PRACTICE SOLUTION First find the scale factor of ∆ JKL to ∆ EFG. 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. = KM KM = 6 6 35 HF 5 5 KM = 42 for Example 5 GUIDED PRACTICE SOLUTION Write proportion. Substitute 35 for HF. Multiply each side by 35 and simplify.

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