Problem:
Semicircle AB has center C and radius 1. Point D is on AB such that CD⊥AB. Extend BD and AD to points E and F, respectively, such that the circular arcs AE and BF have centers at B and A, respectively. Circular arc EF has center D. The area of the shaded "smile" region, denoted by AEFBDA, is
Answer Choices:
A. (2−2​)π
B. 2π−π2​−1
C. (1−22​​)π
D. 25π​−π2​−1
E. (3−22​)π
Solution:
Since CD⊥AB,AC=CB and ∠ADB is inscribed in a semicircle, it follows that △ABD is an isosceles right triangle, ∠BAD=∠ABD=45∘ and BD=2​. Note that ∠EDF=∠ADB=90∘ and DE=BE−BD=2−2​. The area of the "smile" is
