1994 AIME Problems/Problem 2
Problem
A circle with diameter of length 10 is internally tangent at to a circle of radius 20. Square is constructed with and on the larger circle, tangent at to the smaller circle, and the smaller circle outside . The length of can be written in the form , where and are integers. Find .
Note: The diagram was not given during the actual contest.
Solution
Call the center of the larger circle . Extend the diameter to the other side of the square (at point ), and draw . We now have a right triangle, with hypotenuse of length . Since , we know that . The other leg, , is just .
Apply the Pythagorean Theorem:
The quadratic formula shows that the answer is . Discard the negative root, so our answer is .
See also
1994 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 1 |
Followed by Problem 3 | |
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