Question

In this diagram semi-circles are constructed on diameters $AB,$ $AC,$ and $CB,$ so that they are mutually tangent. If $CD\perp AB,$ then the ratio of the shaded area to the area of a circle with $CD$ as radius is:

$1:2$

$1:3$

$\sqrt3:7$

$1:4$

$\sqrt2:6$

Answer: D

Concepts:circle area right triangle area ratio

Difficulty rating: 1990

Small Hint:

Subtract the two small semicircle areas from the large one

Big Hint:

In right triangle $ADB,$ the altitude theorem gives $CD^2=AC\cdot CB$

Solution:

Let $AC=u$ and $CB=v.$ The shaded area is the large semicircle minus the two smaller ones: $\frac{\pi}{8}\bigl((u+v)^2-u^2-v^2\bigr) =\frac{\pi uv}{4}.$ Since $D$ lies on the semicircle with diameter $AB,$ triangle $ADB$ is right, and its altitude satisfies $CD^2=uv.$ A circle of radius $CD$ therefore has area $\pi uv.$ The required ratio is $1:4.$

Therefore, the correct answer is D.

1967 AMC 12 Problem 33

Answer: D

Difficulty rating: 1990

Small Hint:

Big Hint:

Solution:

Problem 33 in Other Years