2006 AMC 12A Problem 13

Problem 13 of 25IntermediateAlgebra Geometry

The vertices of a

3

–

4

–

5

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

r, s, t

are the radii at the vertices, then

r + s = 3,\ r + t = 4,\ s + t = 5.

Adding all three gives

r + s + t = 6,

r = 1,\ s = 2,\ t = 3.

The sum of the areas is

\pi(1^2 + 2^2 + 3^2) = 14\pi.

Thus, the correct answer is E.

Tagged: tangent circles · system of equations · circle area