2008 iTest Problems/Problem 62
Problem
Find the number of values of such that the number of square units in the area of the isosceles triangle with sides
,
, and
is a positive integer.
Solution
Let be the angle between the two sides with length
. The area of the triangle is equal to
.
Note that , so we know that
. Since
is not an integer, for every integer from
to
, there are two possible values of
since
can either be acute or obtuse. Each
corresponds to exactly one value of
so there are
values of
such that the area of the triangle is a positive integer.
See Also
2008 iTest (Problems) | ||
Preceded by: Problem 61 |
Followed by: Problem 63 | |
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 |