Question

Three circles of radius $s$ are drawn in the first quadrant of the $xy$ -plane. The first circle is tangent to both axes, the second is tangent to the first circle and the $x$ -axis, and the third is tangent to the first circle and the $y$ -axis. A circle of radius $r \gt s$ is tangent to both axes and to the second and third circles. What is $r/s?$

$5$

$6$

$8$

$9$

$10$

Solution:

Put the big circle's center at $(r, r)$ and the second small circle's center at $(3s, s).$ They are externally tangent, so the distance between centers is $r + s.$

The horizontal and vertical gaps are $r - 3s$ and $r - s,$ so $(r + s)^2 = (r - 3s)^2 + (r - s)^2.$

Expanding gives $0 = r^2 - 10rs + 9s^2 = (r - 9s)(r - s).$ Since $r \ne s,$ we get $r = 9s,$ so $r/s = 9.$

Thus, the correct answer is D.

2005 AMC 12A Problem 16

Solution:

Problem 16 in Other Years