Given: Given: Given: Given:

A C D B F E G

28. with .

If , , and , find CE. EB = 12 CA = 16 CD = 10 ∠CDE ⬵ ∠B 䉭CDE 䉭CBA

35. Use the result of Exercise 11

to do the following problem. In , and . The length of altitude to side is 6. Find the length of altitude from Q to .

36. Use the result of Exercise 11

to do the following problem. In , and . The length of altitude to side is 5. Find the length of altitude from A to .

37. The distance across a pond is to be measured indirectly by

using similar triangles. If , , , and , find XT. WZ = 50 ft TY = 120 ft YW = 40 ft XY = 160 ft DC AE BC AF BC = 12 AB = 7 ⵥABCD PN QS MN QR QM = 9 QP = 12 ⵥMNPQ Exercises 29, 30 In Exercise 31, provide a two-column proof.

31. Given: ,

Prove: 䉭ABC 䉭EFG BD 7 FG AB 7 DF In Exercise 32, provide a paragraph proof.

32. Given: ,

Prove: 䉭BSR 䉭BCA CB ⬜ AC RS ⬜ AB E F G D N P Q M T U S R Y X Z A B F C D E X Y W Z T Pond Pond Pond A D B C S B A C R

33. Use a two-column proof to prove the following theorem:

“The lengths of the corresponding altitudes of similar triangles have the same ratio as the lengths of any pair of corresponding sides.” Given: ; and are altitudes Prove: DG MQ = DE MN MQ DG 䉭DEF 䉭MNP

34. Provide a paragraph proof for the following problem.

Given: , Prove: RS ZX = ZY RT RU 7 XZ RS 7 YZ Q P N M R S

38. In the figure, . Find AB if and

. DC = 6 AD = 2 ∠ABC ⬵ ∠ADB

39. Prove that the altitude drawn to the hypotenuse of a right

triangle separates the right triangle into two right triangles that are similar to each other and to the original right triangle.

40. Prove that the line segment joining the midpoints of two

sides of a triangle determines a triangle that is similar to the original triangle. see the figure for Exercise 27.

29. with

obtuse angles at vertices D and F as indicated. If , and , find x.

30. with

obtuse angles at vertices D and F. If and : , find . m ∠A 1:3 m ∠CDB = m ∠A m ∠B = 44° 䉭ABF 䉭CBD m ∠AFB = 4x m ∠C = x m ∠B = 45° 䉭ABF 䉭CBD A D E F B C The following theorem, which was proved in Exercise 39 of Section 5.3, will enable us to prove the well-known Pythagorean Theorem. D A B C Figure 5.19 D A B C a Figure 5.18 D A C b D B C c A B C d Pythagorean Theorem Converse of Pythagorean Theorem Pythagorean Triple The Pythagorean Theorem 5.4 KEY CONCEPTS The altitude drawn to the hypotenuse of a right triangle separates the right triangle into two right triangles that are similar to each other and to the original right triangle. THEOREM 5.4.1 Theorem 5.4.1 is illustrated by Figure 5.18, in which the right triangle has its right angle at vertex C so that is the altitude to hypotenuse . The smaller triangles are shown in Figures 5.18b and c, and the original triangle is shown in Figure 5.18d. Note the matching arcs indicating congruent angles. AB CD 䉭ABC Reminder CSSTP means “corresponding sides of similar triangles are proportional.” In Figure 5.18a, and are known as segments parts of the hypotenuse . Furthermore, is the segment of the hypotenuse adjacent to next to leg , and is the segment of the hypotenuse adjacent to leg . Proof of the following theorem is left as an exercise. Compare the statement of Theorem 5.4.2 to the “Prove” statement that follows it. BC BD AC AD AB DB AD The length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the segments of the hypotenuse. THEOREM 5.4.2 GIVEN: in Figure 5.19, with right ; PROVE: PLAN FOR PROOF: Show that . Then use CSSTP. In the proportion , recall that CD is a geometric mean because the second and the third terms are identical. The proof of the following lemma is left as an exercise. Compare the statement of Lemma 5.4.3 to the “Prove” statement that follows it. AD CD = CD DB 䉭ADC 䉭CDB AD CD = CD DB CD ⬜ AB ∠ACB 䉭ABC