If MNPQ is a rhombus, which formula from this section 17.

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