Problem:
Circles centered at A and B each have radius 2, as shown. Point O is the midpoint of AB, and OA=22β. Segments OC and OD are tangent to the circles centered at A and B, respectively, and EF is a common tangent. What is the area of the shaded region ECODF?
Answer Choices:
A. 382ββ
B. 82ββ4βΟ
C. 42β
D. 42β+8Οβ
E. 82ββ2β2Οβ
Solution:
Rectangle ABFE has area AEβ AB=2β 42β=82β. Right triangles ACO and BDO each have hypotenuse 22β and one leg of length 2.
Hence they are each isosceles, and each has area (1/2)(22)=2. Angles CAE and DBF are each 45β, so sectors CAE and DBF each have area
