2005 iTest Problems/Problem 21

Problem

Two circles have a common internal tangent of length $17$ and a common external tangent of length $25$ . Find the product of the radii of the two circles.

Solution 1

The formula for the length of the common external tangent is $\sqrt{d^2-(R-r)^2}$ and the formula for the length of the common internal tangent is $\sqrt{d^2-(R+r)^2}$ , where $d$ is the distance between the centers, and $R$ and $r$ are the radii of the two circles. Given the conditions of the problem, $d^2-(R-r)^2=625$ , $d^2-(R+r)^2=1$ . Subtracting the second equation from the first reveals that $(R+r)^2-(R-r)^2=4Rr=336 \Rightarrow Rr=\boxed{84}$ .

2005 iTest (Problems, Answer Key)
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2005 iTest Problems/Problem 21

Problem

Solution 1

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