is an obtuse triangle with obtuse angle A.

For Exercises 31 and 32, X is the midpoint of and Y is the midpoint of .

31. If , find

.

32. If , find

. A RSTV A RYTX = 13.5 in 2 A RYTX A RSTV = 48 cm 2 TS VT R Y X T S V Exercises 31, 32

33. Given with midpoints

M , N, and P of the sides, explain why . A ABC = 4 A MNP 䉭ABC

35. Given: Square HJKL with LJ ⫽d

Prove: A HJKL = d 2 2

37. Given: The area of right

not shown is 40 in 2 . AC ⫽x BC ⫽x ⫹ 2 Find: x m ∠C = 90° 䉭ABC

36. Given: with

Prove: A RSTV = RS 2 VW ⬵ VT ⵥRSTV In Exercises 34 to 36, provide paragraph proofs.

34. Given: Right

Prove: h = ab c 䉭ABC C M N P A B c B a C A h b L H J K d V T S W R B A D C Exercises 39, 40

38. The lengths of the legs of a right triangle are consecutive

even integers. The numerical value of the area is three times that of the longer leg. Find the lengths of the legs of the triangle.

39. Given: , whose sides are 13 in., 14 in., and 15 in.

Find: a BD, the length of the altitude to the 14-in. side HINT: Use the Pythagorean Theorem twice. b The area of , using the result from part a 䉭ABC 䉭ABC

40. Given: , whose sides are 10 cm, 17 cm, and 21 cm

Find: a BD, the length of the altitude to the 21-cm side b The area of , using the result from part a

41. If the base of a rectangle is increased by 20 percent and

the altitude is increased by 30 percent, by what percentage is the area increased?

42. If the base of a rectangle is increased by 20 percent but

the altitude is decreased by 30 percent, by what percentage is the area changed? Is this an increase or a decrease in area?

43. Given region R ´ S, explain

why . A R ´ S 7 A R 䉭ABC 䉭ABC R S

45. The algebra method of FOIL multiplication is illustrated

geometrically in the drawing. Use the drawing with rectangular regions to complete the following rule: _____________________________ a + bc + d =

44. Given region

, explain why . A R ´ S ´ T = A R + A S + A T R ´ S ´ T R S T a + b c + d b a d c

46. Use the square configuration to complete the following

algebra rule: _________________________ NOTE: Simplify where possible. a + b 2 = b a a b a + b a + b 12 in. 5 in.

56. a Find a lower estimate of the area of the figure by

counting whole squares within the figure. b Find an upper estimate of the area of the figure by counting whole and partial squares within the figure. c Use the average of the results in parts a and b to provide a better estimate of the area of the figure. d Does intuition suggest that the area estimate of part c is the exact answer?

55. a Find a lower estimate of the area of the figure by

counting whole squares within the figure. b Find an upper estimate of the area of the figure by counting whole and partial squares within the figure. c Use the average of the results in parts a and b to provide a better estimate of the area of the figure. d Does intuition suggest that the area estimate of part c is the exact answer?

54. bisects of .

ST ⫽ 6 and TR ⫽ 9. If the area of is 25 m 2 , find the area of . 䉭SVT 䉭RST 䉭STR ∠STR TV

51. The area of a rectangle is 48 in

2 . Where x is the width and y is the length, express the perimeter P of the rectangle in terms only of x.

52. The perimeter of a rectangle is 32 cm. Where x is the

width and y is the length, express the area A of the rectangle in terms only of x.

53. Square DEFG is inscribed in

right as shown. If AD ⫽ 6 and EB ⫽ 8, find the area of square DEFG. 䉭ABC

50. In , AB ⫽

7 and BC ⫽12. The length of altitude to side is 5. Find the length of altitude from A to . DC AE BC AF ⵥABCD

49. In , QP ⫽

12 and QM ⫽9. The length of altitude to side is 6. Find the length of altitude from Q to . PN QS MN QR ⵥMNPQ

48. In the triangle whose sides are 13, 20, and 21 cm long, the

length of the altitude drawn to the 21-cm side is 12 cm. Find the lengths of the remaining altitudes of the triangle. In Exercises 47 to 50, use the fact that the area of the polygon is unique.

47. In the right triangle, find the length of the altitude drawn

to the hypotenuse. 20 cm 21 cm 13 cm 12 cm Q P N M R S A B F C D E A B D E G F C R T V S