2005 AMC 12A Problem 16

Problem 16 of 25IntermediateGeometry

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

\frac{r}{s}?

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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,

(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,

\frac{r}{s} = 9.

Thus, the correct answer is D.