1995 AIME Problems/Problem 4
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 1
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
Solution 2 (Analytic Geometry)
Let be defined as the origin of a coordinate plane with the -axis running across the chord and by the Pythagorean Theorem. Then we have and , and since , the point is one-third of the way from to , so point has coordinates . is the center of the circle with radius , so the equation of this circle is . Since the chord's equation is , we must find all values of satisfying the equation of the circle such that . We find that , so the chord has length and the answer is .
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See also
1995 AIME (Problems • Answer Key • Resources) | ||
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Followed by Problem 5 | |
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