Difference between revisions of "2002 AIME II Problems/Problem 14"
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Revision as of 20:37, 4 July 2013
Problem
The perimeter of triangle is
, and the angle
is a right angle. A circle of radius
with center
on
is drawn so that it is tangent to
and
. Given that
where
and
are relatively prime positive integers, find
.
Solution
Let the circle intersect at
. Then note
and
are similar. Also note that
by power of a point. So we have
Solving,
. So the ratio of the side lengths of the triangles is 2. Therefore,
so
and
Substituting for
, we see that
, so
and the answer is
.
See also
2002 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.