1997 AIME Problems/Problem 4

Problem

Circles of radii and are mutually externally tangent, where and are relatively prime positive integers. Find

Solution

If (in the diagram above) we draw the line going through the centers of the circles with radii and , that line is the perpendicular bisector of the segment connecting the centers of the two circles with radii . Then we form two right triangles, of lengths and , wher is the distance between the center of the circle in question and the segment connecting the centers of the two circles of radii . By the Pythagorean Theorem, we now have two equations with two unknowns:

$\begin{eqnarray*} 5^2 + x^2 &=& (5+r)^2 \\ x &=& \sqrt{10r + r^2} \\ && \\ (8 + r + \sqrt{10r+r^2})^2 + 5^2 &=& 13^2\\ 8 + r + \sqrt{10r+r^2} &=& 12\\ \sqrt{10r+r^2}&=& 4-r\\ 10r+r^2 &=& 16 - 8r + r^2\\ r &=& \frac{8}{9} \end{eqnarray*}$ (Error compiling LaTeX. Unknown error_msg)

So .

See also

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.