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 $\boxed{196}\pi$.

See Also

Mock AIME 3 Pre 2005 (Problems, Source)
Preceded by
First Question
Followed by
Problem 2
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