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2006 AMC 12A Problem 13

Problem 13 of 25IntermediateAlgebraGeometry

The vertices of a 33–44–55 right triangle are the centers of three mutually externally tangent circles, as shown. What is the sum of the areas of these circles?

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Solution

If r,s,tr, s, t are the radii at the vertices, then r+s=3, r+t=4, s+t=5.r + s = 3,\ r + t = 4,\ s + t = 5. Adding all three gives r+s+t=6,r + s + t = 6, so r=1, s=2, t=3.r = 1,\ s = 2,\ t = 3. The sum of the areas is π(12+22+32)=14π.\pi(1^2 + 2^2 + 3^2) = 14\pi. Thus, the correct answer is E.

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Concepts: tangent circles · system of equations · circle area

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.