1992 AMC 12 Problem 11

Question

The ratio of the radii of two concentric circles is

1:3.

If

\overline{AC}

is a diameter of the larger circle,

\overline{BC}

is a chord of the larger circle that is tangent to the smaller circle, and

AB=12,

then the radius of the larger circle is

13

18

21

24

26

Draw the radius to the point where

\overline{BC}

touches the smaller circle

Use the similar right triangles formed by the tangent radius and the diameter

Let

O

be the common center and

T

the tangency point on

\overline{BC}.

Since

OT\perp BC

and

\angle ABC=90^\circ,

triangles

OTC

and

ABC

are similar. If the larger radius is

R,

then

OT=\frac{R}{3},

OC=R,

and

AC=2R.

Thus

\frac{OT}{AB}=\frac{OC}{AC}=\frac12.

Hence

OT=6

and

R=3OT=18.

Thus the correct answer is B.

Problem 11 in other years