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Warm Up

Use Similar Polygons. Warm Up. Lesson Presentation. Lesson Quiz. 12. x. 1. Solve. =. 10. 60. 72. ANSWER. 0.43 cm. ANSWER. Warm-Up. 2. The scale of a map is 1 cm : 10 mi . The actual distance between two towns is 4.3 miles. Find the length on the map. 1 cm : 2 m. ANSWER.

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Warm Up

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  1. Use Similar Polygons Warm Up Lesson Presentation Lesson Quiz

  2. 12 x 1.Solve = 10 60 72 ANSWER 0.43 cm ANSWER Warm-Up 2.The scale of a map is 1cm:10mi. The actual distance between two towns is 4.3 miles. Find the length on the map.

  3. 1 cm : 2 m ANSWER Warm-Up 3.A model train engine is 9 centimeters long. The actual engine is 18 meters long. What is the scale of the model?

  4. a. ~ ~ ~ R X, T Z and Y S = = = 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. SOLUTION

  5. 5 ST 30 25 TR 5 RS 5 20 b. ; ; = = = = = = 3 15 3 ZX 3 18 XY 12 YZ 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. SOLUTION

  6. . c. Because the ratios in part (b) are equal, TR ST RS = = YZ ZX XY 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. SOLUTION

  7. ANSWER ; ~ ~ ~ J P, L R and Q K = = = JK LJ KL = = RP PQ QR Guided Practice 1. Given ∆ JKL ~ ∆ PQR, list all pairs of congruent angles. Write the ratios of the corresponding side lengths in a statement of proportionality.

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

  9. 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 angles, so W J.So,the corresponding angles are congruent. Example 2 SOLUTION

  10. 5 5 5 5 = = = = WZ 20 15 XW 25 ZY 30 YX 4 4 4 4 20 GH HJ 24 12 JF FG 16 STEP 2 Show that corresponding side lengths are proportional. = = = = Example 2 SOLUTION

  11. 5 4 . So ZYXW ~ FGHJ. The scale factor of ZYXWto FGHJis Example 2 SOLUTION The ratios are equal, so the corresponding side lengths are proportional.

  12. In the diagram, ∆DEF ~ ∆MNP. Find the value of x. ALGEBRA = NP MN EF DE 20 12 = x 9 Example 3 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.

  13. In the diagram, ABCD ~ QRST. 1 ANSWER 2 8 ANSWER Guided Practice 2. What is the scale factor of QRSTto ABCD ? 3. Find the value of x.

  14. b. Find the perimeter of an Olympic pool and the new pool. 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

  15. 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 = a. Because the new pool will be similar to an Olympic pool, the scale factor is the ratio of the lengths, 150 4 4 40 5 5 50 = Example 4 SOLUTION 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.

  16. In the diagram, ABCDE ~ FGHJK. 3 ANSWER 2 12 46 ANSWER ANSWER Guided Practice 4. Find the scale factor of FGHJKto ABCDE. 5. Find the value of x. 6. Find The perimeter ofABCDE.

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

  18. = PS 3 3 4 PY 4 PS = 20 = PS 15 The length of the altitude PSis 15. Example 5 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.

  19. 7. In the diagram, ∆JKL ~ ∆ EFG. Find the length of the median KM. 42 ANSWER Guided Practice

  20. Yes; EFGH ~KLMN; the scale factor is 2:1 ANSWER Lesson Quiz 1. Determine whether the polygons are similar. If they are, write a similarity statement and find the scale factor of EFGH to KLMN.

  21. 2.Inthe diagram,DEF ~ HJK.Find the value of x. ANSWER 13.5 Lesson Quiz

  22. ANSWER 5 : 4 ; 5 : 4 ANSWER 3 : 7; 9: 49 Lesson Quiz 3. Two similar triangles have the scale factor 5 : 4. Find the ratio of their corresponding altitudes and median. 4. Two similar triangles have the scale factor 3 : 7. Find the ratio of their corresponding perimeters and areas.

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