Mock AIME 3 Pre 2005 Problems/Problem 1

Problem

Three circles are mutually externally tangent. Two of the circles have radii $3$ and $7$ . If the area of the triangle formed by connecting their centers is $84$ , then the area of the third circle is $k\pi$ for some integer $k$ . Determine $k$ .

Solution

Let the radius of the third circle be $r$ . The side lengths of the triangle are $10$ , $3+r$ , and $7+r$ . From Heron's Formula, $84=\sqrt{(10+r)(r)(7)(3)}$ , or $84*84=r(10+r)*21$ , or $84*4=r(10+r)$ . $84*4=14*24$ , so $r=14$ . Thus the area of the circle is $\boxed{196}\pi$ .

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Mock AIME 3 Pre 2005 Problems/Problem 1

Problem

Solution

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