2006 AMC 10B Problem 19

Problem 19 of 25HarderGeometry

A circle of radius

2

is centered at

O.

Square

OABC

has side length

1.

Sides

\overline{AB}

and

\overline{CB}

are extended past

B

to meet the circle at

D

and

E,

respectively. What is the area of the shaded region in the figure, which is bounded by

\overline{BD},

\overline{BE},

and the minor arc connecting

D

and

E?

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Solution

Since

OA=1

and

OD=2

with

D

on the line

x=1,

we get

\angle AOD=60^\circ,

and likewise

\angle COE=60^\circ,

\angle DOE=30^\circ.

The sector

DOE

has area

\tfrac{30}{360}\pi(2^2)=\tfrac{\pi}{3}.

The region is this sector minus triangles

OBD

and

OBE.

With

BD=BE=\sqrt3-1,

each triangle has area

\tfrac12(\sqrt3-1)(1),

totaling

\sqrt3-1.

So the shaded area is

\tfrac{\pi}{3}-(\sqrt3-1)=\tfrac{\pi}{3}+1-\sqrt3.

Thus, the correct answer is A.