2005 AMC 12A Problem 16

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

\frac{r}{s}?

5

6

8

9

10

Place the centers by their radii: the big circle is at

(r, r)

and the second small circle at

(3s, s)

The distance between the big center and a tangent small center equals

r + s;

apply the Pythagorean theorem to the horizontal and vertical gaps

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

\frac{r}{s} = 9.

Thus, the correct answer is D.

Problem 16 in other years