2011 AMC 12A Problem 11

Question

Circles

A,

B,

and

C

each have radius

1.

Circles

A

and

B

share one point of tangency. Circle

C

has a point of tangency with the midpoint of

\overline{AB}.

What is the area inside circle

C

but outside circle

A

and circle

B?

3 - \dfrac{\pi}{2}

\dfrac{\pi}{2}

2

\dfrac{3\pi}{4}

1 + \dfrac{\pi}{2}

Place the centers at

A(-1,0),

B(1,0),

and

C(0,1);

the distance from

C

to each of

A, B

is

\sqrt2

The wanted area is the area of

C

minus the two lens-shaped overlaps of

C

with

A

and with

B

Place

A = (-1, 0),

B = (1, 0),

so their tangency point is the origin, the midpoint of

\overline{AB}.

Then

C = (0, 1),

since

C

passes through the origin.

The distance from

C

to

A

(and to

B

) is

\sqrt2.

Two unit circles whose centers are

\sqrt2

apart overlap in a lens of area

\begin{aligned} 2\cos^{-1}\!\left(\tfrac{\sqrt2}{2}\right) \\ {}- \tfrac{\sqrt2}{2}\sqrt{4 - 2} \\ = 2 \cdot \tfrac{\pi}{4} - 1 = \tfrac{\pi}{2} - 1. \end{aligned}

Circles

A

and

B

meet only at the origin, so the two lenses do not overlap. The wanted area is

\pi - 2\left(\tfrac{\pi}{2} - 1\right) = 2.

Thus, the correct answer is C.

Problem 11 in other years