2016 AIME II Problems/Problem 5
Triangle has a right angle at
. Its side lengths are pariwise relatively prime positive integers, and its perimeter is
. Let
be the foot of the altitude to
, and for
, let
be the foot of the altitude to
in
. The sum
. Find
.
Solution
Note that by counting the area in 2 ways, the first altitude is . By similar triangles, the common ratio is
for reach height, so by the geometric series formula, we have
. Multiplying by the denominator and expanding, the equation becomes
. Cancelling
and multiplying by
yields
, so
and
. Checking for Pythagorean triples gives
and
, so
Solution modified/fixed from Shaddoll's solution.