Difference between revisions of "1995 AIME Problems/Problem 4"
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Latest revision as of 19:29, 4 July 2013
Problem
Circles of radius and are externally tangent to each other and are internally tangent to a circle of radius . The circle of radius has a chord that is a common external tangent of the other two circles. Find the square of the length of this chord.
Solution
We label the points as following: the centers of the circles of radii are respectively, and the endpoints of the chord are . Let be the feet of the perpendiculars from to (so are the points of tangency). Then we note that , and . Thus, (consider similar triangles). Applying the Pythagorean Theorem to , we find that
See also
1995 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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