2007 AMC 10A Problem 24

Problem 24 of 25HarderGeometry

Circles centered at

A

and

B

each have radius

2,

as shown. Point

O

is the midpoint of

\overline{AB},

and

OA = 2\sqrt{2}.

Segments

OC

and

OD

are tangent to the circles centered at

A

and

B,

respectively, and

\overline{EF}

is a common tangent. What is the area of the shaded region

ECODF?

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Solution

Rectangle

ABFE

has area

AE \cdot AB = 2 \cdot 4\sqrt2 = 8\sqrt2.

Right triangles

ACO

and

BDO

each have hypotenuse

2\sqrt2

and a leg of

2,

so each is isosceles right with area

2.

Angles

CAE

and

DBF

are each

45^\circ,

so sectors

CAE

and

DBF

each have area

\tfrac18 \pi \cdot 2^2 = \tfrac{\pi}{2}.

The shaded area is

\begin{gathered} 8\sqrt2 - 2 \cdot 2 - 2 \cdot \tfrac{\pi}{2} \\ = 8\sqrt2 - 4 - \pi. \end{gathered}

Thus, the correct answer is B.