Question

Diameter $ACE$ is divided at $C$ in the ratio $2:3.$ The two semicircles, $ABC$ and $CDE,$ divide the circular region into an upper (shaded) region and a lower region. The ratio of the area of the upper region to that of the lower region is

$2:3$

$1:1$

$3:2$

$9:4$

$5:2$

Answer: C

Concepts:circle area area decomposition

Difficulty rating: 1610

Solution:

Choose $AE=10$ so that $AC=4$ and $CE=6.$ The large semicircle has radius $5,$ so its area is $\frac{25\pi}{2}.$

The semicircle on $AC$ has radius $2$ and area $2\pi,$ while the semicircle on $CE$ has radius $3$ and area $\frac{9\pi}{2}.$

The upper shaded region has area $\frac{25\pi}{2}-2\pi+\frac{9\pi}{2}=15\pi.$

The whole circle has area $25\pi,$ so the lower region has area $25\pi-15\pi=10\pi.$

The desired ratio is $15\pi:10\pi=3:2.$

Thus, C is the correct answer.

1997 AMC 8 Problem 24

Answer: C

Difficulty rating: 1610

Solution:

Problem 24 in Other Years