2010 AMC 8 Problem 23

Question

Semicircles

POQ

and

ROS

pass through the center of circle

O.

What is the ratio of the combined areas of the two semicircles to the area of circle

O?

\dfrac{\sqrt 2}{4}

\dfrac{1}{2}

\dfrac{2}{\pi}

\dfrac{2}{3}

\dfrac{\sqrt 2}{2}

The two semicircles together have the area of one circle of radius

1

.

Find the radius of circle

O

from the coordinates.

The area of each of the semicircles is

\pi \dfrac {r^2}{2} .

Each of them has a radius of

1,

so their combined area is

\pi \frac{1}{2} + \pi \frac 12 = \pi.

Next, the radius of the larger circle is equal to the length of

OQ,

which is equal to

\sqrt{1^2+1^2} = \sqrt 2.

Its area is

\pi r^2 = \pi(\sqrt{2})^2 = 2\pi.

This means the ratio is

\dfrac{\pi}{2\pi} = \frac 12.

Thus, the answer is B .

Problem 23 in other years