1993 AIME Problems/Problem 15
Problem
Let be an altitude of
. Let
and
be the points where the circles inscribed in the triangles
and
are tangent to
. If
,
, and
, then
can be expressed as
, where
and
are relatively prime integers. Find
.
Solution
From the Pythagorean Theorem, , and
. Subtracting those two equations yields
. After simplification, we see that
, or
. Note that
. Therefore we have that
. Therefore
.
Now note that ,
, and
. Therefore we have
.
Plugging in and simplifying, we have
.
Edit by GameMaster402: It can be shown that in any triangle with side lengths , if you draw an altitude from the vertex to the side of
, and draw the incircles of the two right triangles, the distance between the two tangency points is simply
See also
1993 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 14 |
Followed by Last question | |
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.