Problem:
Circles centered at A and B each have radius 2, as shown. Point O is the midpoint of AB, and OA=22​. 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. 382​​
B. 82​−4−π
C. 42​
D. 42​+8π​
E. 82​−2−2π​
Solution:
Rectangle ABFE has area AE⋅AB=2⋅42​=82​. Right triangles ACO and BDO each have hypotenuse 22​ 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 45∘, so sectors CAE and DBF each have area
