2000 AMC 12 Problem 24

Question

If circular arcs

AC

and

BC

have centers at

B

and

A,

respectively, then there exists a circle tangent to both arc

AC

and arc

BC,

and to

\overline{AB}.

If the length of arc

BC

is

12,

then what is the circumference of the circle?

24

25

26

27

28

Answer: D

Concepts:power of a point tangent circles arc

Difficulty rating: 2390

Small hint:

Both arcs have radius

AB,

and

C

is at distance

AB

from both

A

and

B,

so

\triangle ABC

is equilateral

Big hint:

Use Power of a Point at

A

with

AD = \tfrac12 AB

to find the small circle’s radius

r = \tfrac38 AB

Solution:

Each arc has radius

AB,

and

C

is at distance

AB

from both

A

and

B,

so

\triangle ABC

is equilateral. Thus arc

BC

subtends

60^\circ

of a circle of radius

AB,

whose full circumference is

6 \cdot 12 = 72.

Let the small circle have radius

r

and touch

\overline{AB}

at its midpoint

D,

where

AD = \tfrac12 AB.

By Power of a Point,

AD^2 = AB(AB - 2r),

so

\frac{AB^2}{4} = AB^2 - 2r\,AB,

giving

2r = \tfrac34 AB,

hence

r = \tfrac38 AB.

The circumferences are in the ratio of the radii, so the small circle’s circumference is

\tfrac38 \cdot 72 = 27.

Thus, the correct answer is D.