Problem: In this diagram semi-circles are constructed on diameters AB,AC, and CB, so that they are mutually tangent. If CDβ₯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. 3β:7
D. 1:4
E. 2β:6
Solution:
Shaded area =21β(4ΟβAB2β4ΟβAC2β4ΟβCB2). Since AB=AC+CB, shaded area =21ββ 4Οβ(AC2+2ACβ CB+CB2βAC2βCB2)=4Οβ(AC)(CB). But the area of the required circle equals ΟCD2, and since CD2=(AC)(CB), the area of the circle equals Ο(AC)(CB). Therefore, the required ratio is 1:4.
