2008 iTest Problems/Problem 37
Contents
[hide]Problem
A triangle has sides of length , and . Let a and b be relatively prime positive integers such that is the length of the shortest altitude of the triangle. Find the value of .
Solutions
Solution 1
Note that , so the side lengths are part of a right triangle. Also note that the shortest altitude is the one that is perpendicular to the longest side. Let be the shortest altitude. Using the area formula, Thus, .
Solution 2
Using Heron's Formula, the area of the triangle is Note that the shortest altitude is the one that is perpendicular to the longest side. Let be the shortest altitude. Using the area formula, Thus, .
See Also
2008 iTest (Problems) | ||
Preceded by: Problem 36 |
Followed by: Problem 38 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 • 51 • 52 • 53 • 54 • 55 • 56 • 57 • 58 • 59 • 60 • 61 • 62 • 63 • 64 • 65 • 66 • 67 • 68 • 69 • 70 • 71 • 72 • 73 • 74 • 75 • 76 • 77 • 78 • 79 • 80 • 81 • 82 • 83 • 84 • 85 • 86 • 87 • 88 • 89 • 90 • 91 • 92 • 93 • 94 • 95 • 96 • 97 • 98 • 99 • 100 |