1994 AIME Problems/Problem 10
Contents
Problem
In triangle angle
is a right angle and the altitude from
meets
at
The lengths of the sides of
are integers,
and
, where
and
are relatively prime positive integers. Find
Solution 1
Since , we have
. It follows that
and
, so
and
are in the form
and
, respectively.
By the Pythagorean Theorem, we find that , so
. Letting
, we obtain after dividing through by
,
. As
, the pairs of factors of
are
; clearly
, so
. Then,
.
Thus, , and
.
Solution 2
We will solve for using
, which gives us
. By the Pythagorean Theorem on
, we have
. Trying out factors of
, we can either guess and check or just guess to find that
and
(The other pairs give answers over 999). Adding these, we have
and
, and our answer is
.
See also
1994 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 9 |
Followed by Problem 11 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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