Warm Up Solve each proportion. 1. 2. 3.

1. Warm Up Solve each proportion. 1.2. 3. AB = 16 QR = 10.5 x = 21

2. Applying Properties of Similar Triangles 7-4 Warm Up Lesson Presentation Lesson Quiz Holt Geometry

3. Objectives Use properties of similar triangles to find segment lengths. Apply proportionality and triangle angle bisector theorems.

4. Artists use mathematical techniques to make two-dimensional paintings appear three-dimensional. The invention of perspective was based on the observation that far away objects look smaller and closer objects look larger. Mathematical theorems like the Triangle Proportionality Theorem are important in making perspective drawings.

6. It is given that , so by the Triangle Proportionality Theorem. Example 1: Finding the Length of a Segment Find US. Substitute 14 for RU, 4 for VT, and 10 for RV. Cross Products Prop. US(10) = 56 Divide both sides by 10.

7. Check It Out! Example 1 Find PN. Use the Triangle Proportionality Theorem. Substitute in the given values. Cross Products Prop. 2PN = 15 PN = 7.5 Divide both sides by 2.

9. Verify that . Since , by the Converse of the Triangle Proportionality Theorem. Example 2: Verifying Segments are Parallel

10. AC = 36 cm, and BC = 27 cm. Verify that . Since , by the Converse of the Triangle Proportionality Theorem. Check It Out! Example 2

12. The previous theorems and corollary lead to the following conclusion.

13. by the ∆ Bisector Theorem. Example 4: Using the Triangle Angle Bisector Theorem Find PS and SR. Substitute the given values. 40(x – 2) = 32(x + 5) Cross Products Property 40x – 80 = 32x + 160 Distributive Property

14. Example 4 Continued 40x – 80 = 32x + 160 8x = 240 Simplify. x = 30 Divide both sides by 8. Substitute 30 for x. PS = x – 2 SR = x + 5 = 30 – 2 = 28 = 30 + 5 = 35

15. by the ∆ Bisector Theorem. Check It Out! Example 4 Find AC and DC. Substitute in given values. 4y = 4.5y – 9 Cross Products Theorem –0.5y = –9 Simplify. y = 18 Divide both sides by –0.5. So DC = 9 and AC = 16.

17. Lesson Quiz: Part I Find the length of each segment. 1.2. SR = 25, ST = 15

18. 3.Verify that BE and CD are parallel. Since , by the Converse of the ∆Proportionality Thm. Lesson Quiz: Part II

19. 7-5 Using Proportional Relationships Warm Up Lesson Presentation Lesson Quiz Holt Geometry

20. Vocabulary indirect measurement scale drawing scale

21. Indirect measurementis any method that uses formulas, similar figures, and/or proportions to measure an object. The following example shows one indirect measurement technique.

22. Helpful Hint Whenever dimensions are given in both feet and inches, you must convert them to either feet or inches before doing any calculations.

23. Example 1: Measurement Application Tyler wants to find the height of a telephone pole. He measured the pole’s shadow and his own shadow and then made a diagram. What is the height h of the pole?

24. Example 1 Continued Step 1 Convert the measurements to inches. AB = 7 ft 8 in. = (7  12) in. + 8 in. = 92 in. BC = 5 ft 9 in. = (5  12) in. + 9 in. = 69 in. FG = 38 ft 4 in. = (38  12) in. + 4 in. = 460 in. Step 2 Find similar triangles. Because the sun’s rays are parallel, A  F. Therefore ∆ABC ~ ∆FGH by AA ~.

25. Example 1 Continued Step 3 Find h. Corr. sides are proportional. Substitute 69 for BC, h for GH, 92 for AB, and 460 for FG. Cross Products Prop. 92h = 69  460 Divide both sides by 92. h = 345 The height h of the pole is 345 inches, or 28 feet 9 inches.

26. Check It Out! Example 1 A student who is 5 ft 6 in. tall measured shadows to find the height LM of a flagpole. What is LM? Step 1 Convert the measurements to inches. GH = 5 ft 6 in. = (5  12) in. + 6 in. = 66 in. JH = 5 ft = (5  12) in. = 60 in. NM = 14 ft 2 in. = (14  12) in. + 2 in. = 170 in.

27. Check It Out! Example 1 Continued Step 2 Find similar triangles. Because the sun’s rays are parallel, L  G. Therefore ∆JGH ~ ∆NLM by AA ~. Step 3 Find h. Corr. sides are proportional. Substitute 66 for BC, h for LM, 60 for JH, and 170 for MN. Cross Products Prop. 60(h) = 66  170 h = 187 Divide both sides by 60. The height of the flagpole is 187 in., or 15 ft. 7 in.

28. A scale drawingrepresents an object as smaller than or larger than its actual size. The drawing’s scaleis the ratio of any length in the drawing to the corresponding actual length. For example, on a map with a scale of 1 cm : 1500 m, one centimeter on the map represents 1500 m in actual distance.

29. Remember! A proportion may compare measurements that have different units.

30. Check It Out! Example 3 The rectangular central chamber of the Lincoln Memorial is 74 ft long and 60 ft wide. Make a scale drawing of the floor of the chamber using a scale of 1 in.:20 ft.

31. 3.7 in. 3 in. Check It Out! Example 3 Continued Set up proportions to find the length l and width w of the scale drawing. 20w = 60 w = 3 in

34. The similarity ratio of ∆LMN to ∆QRS is By the Proportional Perimeters and Areas Theorem, the ratio of the triangles’ perimeters is also , and the ratio of the triangles’ areas is Example 4: Using Ratios to Find Perimeters and Areas Given that ∆LMN:∆QRT, find the perimeter P and area A of ∆QRS.

35. Example 4 Continued Perimeter Area 13P = 36(9.1) 132A = (9.1)2(60) P = 25.2 A = 29.4 cm2 The perimeter of ∆QRS is 25.2 cm, and the area is 29.4 cm2.

36. Check It Out! Example 4 ∆ABC ~ ∆DEF, BC = 4 mm, and EF = 12 mm. If P = 42 mm and A = 96 mm2 for ∆DEF, find the perimeter and area of ∆ABC. Perimeter Area 12P = 42(4) 122A = (4)2(96) P = 14 mm The perimeter of ∆ABC is 14 mm, and the area is 10.7 mm2.

38. Lesson Quiz: Part I 1. Maria is 4 ft 2 in. tall. To find the height of a flagpole, she measured her shadow and the pole’s shadow. What is the height h of the flagpole? 2. A blueprint for Latisha’s bedroom uses a scale of 1 in.:4 ft. Her bedroom on the blueprint is 3 in. long. How long is the actual room? 25 ft 12 ft

39. Lesson Quiz: Part II 3. ∆ABC ~ ∆DEF. Find the perimeter and area of ∆ABC. P = 27 in., A = 31.5 in2

40. Dilations and Similarity in the Coordinate Plane 7-6 Warm Up Lesson Presentation Lesson Quiz Holt Geometry

41. Warm Up Simplify each radical. 1.2.3. Find the distance between each pair of points. Write your answer in simplest radical form. 4. C (1, 6) and D (–2, 0) 5.E(–7, –1) and F(–1, –5)

42. Objectives Apply similarity properties in the coordinate plane. Use coordinate proof to prove figures similar.

43. Vocabulary dilation scale factor

44. Helpful Hint If the scale factor of a dilation is greater than 1 (k > 1), it is an enlargement. If the scale factor is less than 1 (k < 1), it is a reduction.

45. Draw the border of the photo after a dilation with scale factor Example 1: Computer Graphics Application

46. Step 1 Multiply the vertices of the photo A(0, 0), B(0, 4), C(3, 4), and D(3, 0) by Example 1 Continued Rectangle ABCD Rectangle A’B’C’D’

47. Example 1 Continued Step 2 Plot points A’(0, 0), B’(0, 10), C’(7.5, 10), and D’(7.5, 0). Draw the rectangle.

48. What if…?Draw the border of the original photo after a dilation with scale factor Check It Out! Example 1

49. Step 1 Multiply the vertices of the photo A(0, 0), B(0, 4), C(3, 4), and D(3, 0) by Check It Out! Example 1 Continued Rectangle ABCD Rectangle A’B’C’D’

50. B’ C’ 2 0 1.5 A’ D’ Check It Out! Example 1 Continued Step 2 Plot points A’(0, 0), B’(0, 2), C’(1.5, 2), and D’(1.5, 0). Draw the rectangle.