2004 AMC 10A Problem 21

Problem 21 of 25HarderAlgebra Geometry

Two distinct lines pass through the center of three concentric circles of radii

3,

2,

and

1.

The area of the shaded region in the diagram is

\dfrac{8}{13}

of the area of the unshaded region. What is the radian measure of the acute angle formed by the two lines? (Note:

\pi

radians is

180

degrees.)

Show solution

Solution

Let

\theta

be the acute angle. The shaded region has three parts: two acute sectors of the unit disk with total area

\theta,

two obtuse sectors of the ring between radii

1

and

2

with total area

3(\pi - \theta),

and two acute sectors of the ring between radii

2

and

3

with total area

5\theta.

Adding these gives a shaded area of

\theta + 3(\pi - \theta) + 5\theta = 3\pi + 3\theta.

The shaded region is

\dfrac{8}{13}

of the unshaded region, so it is

\dfrac{8}{21}

of the total area

9\pi.

Then

3\pi + 3\theta = \dfrac{8}{21}(9\pi) = \dfrac{24\pi}{7},

which gives

\theta = \dfrac{\pi}{7}.

Thus, the correct answer is B.