1998 AMC 12 Problem 16

Question

The figure shown is the union of a circle and two semicircles of diameters

a

and

b,

all of whose centers are collinear. The ratio of the area of the shaded region to that of the unshaded region is

\sqrt{\frac ab}

\frac ab

\frac{a^2}{b^2}

\frac{a+b}{2b}

\frac{a^2+2ab}{b^2+2ab}

Express each region using the large semicircle and one small semicircle

Both resulting areas factor a common multiple of

a+b

The shaded area is

\begin{aligned} &\frac{\pi}{2}\left(\frac{a+b}{2}\right)^2 +\frac{\pi}{2}\left(\frac a2\right)^2\\ &\qquad-\frac{\pi}{2}\left(\frac b2\right)^2 =\frac{\pi a(a+b)}4. \end{aligned}

Similarly, the unshaded area is

\frac{\pi b(a+b)}{4}.

Their ratio is

\frac{a}{b},

so B is correct.

Problem 16 in other years