2007 AIME I Problems/Problem 9
Problem
In right triangle with right angle
,
and
. Its legs
and
are extended beyond
and
. Points
and
lie in the exterior of the triangle and are the centers of two circles with equal radii. The circle with center
is tangent to the hypotenuse and to the extension of leg
, the circle with center
is tangent to the hypotenuse and to the extension of leg
, and the circles are externally tangent to each other. The length of the radius either circle can be expressed as
, where
and
are relatively prime positive integers. Find
.
Solution
Let the point where CB's extension hits the circle be D, and the point where the hypotenuse hits that circle be E. Clearly . Let
. Draw the two perpendicular radii to D and E. Now we have a cyclic quadrilateral. Let the radius be length
. We see that since the cosine of angle ABC is
the cosine of angle EBD is
. Since the measure of the angle opposite to EBD is the complement of this one, its cosine is
. Using the law of cosines, we see that
This tells us that
.
Now look at the other end of the hypotenuse. Call the point where CA hits the circle F and the point where the hypotenuse hits the circle G. Draw the radii to F and G and we have cyclic quadrilaterals once more.
Using the law of cosines again, we find that the length of our tangents is . Note that if we connect the centers of the circles we have a rectangle with sidelengths 8x and 4x. So, 8x+2.4x+x=34. Solving we find that
so our answer is 737.
See also
2007 AIME I (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 |