2007 AIME II Problems/Problem 9
Problem
Rectangle is given with
and
Points
and
lie on
and
respectively, such that
The inscribed circle of triangle
is tangent to
at point
and the inscribed circle of triangle
is tangent to
at point
Find
Solution
Solution 1
Several Pythagorean triples exist amongst the numbers given. . Also, the length of
.
Use the Two Tangent theorem on . Since both circles are inscribed in congruent triangles, they are congruent; therefore,
. By the Two Tangent theorem, note that
, making
. Also,
.
.
Finally, . Also,
. Equating, we see that
, so
.
Solution 2
By the Two Tangent theorem, we have that . Solve for
. Also,
, so
. Since
, this can become
. Substituting in their values, the answer is
.
See also
2007 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 8 |
Followed by Problem 10 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |