Problem: In this diagram semi-circles are constructed on diameters AB,AC, and CB, so that they are mutually tangent. If CD⊥AB, then the ratio of the shaded area to the area of a circle with CD as radius is:
Answer Choices:
A. 1:2
B. 1:3
C. 3​:7
D. 1:4
E. 2​:6
Solution:
Shaded area =21​(4π​AB2−4π​AC2−4π​CB2). Since AB=AC+CB, shaded area =21​⋅4π​(AC2+2AC⋅CB+CB2−AC2−CB2)=4π​(AC)(CB). But the area of the required circle equals πCD2, and since CD2=(AC)(CB), the area of the circle equals π(AC)(CB). Therefore, the required ratio is 1:4.
