Determine whether each triple a, b, c is a Pythagorean Determine the type of triangle represented if the lengths

38. When the rectangle in the accompanying drawing whose

dimensions are 16 by 9 is cut into pieces and rearranged, a square can be formed. What is the perimeter of this square?

42. Prove that if a, b, c is a Pythagorean triple and n is a

natural number, then na, nb, nc is also a Pythagorean triple.

43. Use Figure 5.19 to prove Theorem 5.4.2. 44. Use Figures 5.20 and 5.21 to prove Lemma 5.4.3.

40. Find the length of the altitude to the 8-in. side of a

triangle whose sides are 4, 6, and 8 in. long. HINT: See Example 5.

41. In the figure, square RSTV has its vertices on the sides of

square WXYZ as shown. If and , find TS. Also find RT. TY = 12 ZT = 5

39. A, C, and F are three of the vertices of the cube shown in

the accompanying figure. Given that each face of the cube is a square, what is the measure of angle ACF? 9 5 10 3 16 C F A W R V S X Z Y T The 45°-45°-90° Triangle The 30°-60°-90° Triangle Special Right Triangles 5.5 KEY CONCEPTS Many of the calculations that we do in this section involve square root radicals. To un- derstand some of these calculations better, it may be necessary to review the Properties of Square Roots in Appendix A.4. Certain right triangles occur so often that they deserve more attention than others. The two special right triangles that we consider in this section have angle measures of 45°, 45°, and 90° or of 30°, 60°, and 90°. THE 45°-45°-90° RIGHT TRIANGLE In the 45º-45º-90º triangle, the legs are opposite the congruent angles and are also con- gruent. Rather than using a and b to represent the lengths of the legs, we use a for both lengths, as shown in Figure 5.31. By the Pythagorean Theorem, it follows that c = a 12 c = 12 2a 2 c = 22a 2 c 2 = 2a 2 c 2 = a 2 + a 2 In a triangle whose angles measure 45°, 45°, and 90°, the hypotenuse has a length equal to the product of and the length of either leg. 12 THEOREM 5.5.1 왘 45-45-90 Theorem a 45° 45° a ? Figure 5.31 a 45° 45° a a 2 Figure 5.32 It is better to memorize the sketch in Figure 5.32 than to repeat the steps of the “proof ” that precedes the 45-45-90 Theorem. Exs. 1–3 EXAMPLE 1 Find the lengths of the missing sides in each triangle in Figure 5.33. Reminder If two angles of a triangle are congruent, then the sides opposite