Problem:
Semicircle AB has center C and radius 1. Point D is on AB such that CDβ₯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. (2β2β)Ο
B. 2ΟβΟ2ββ1
C. (1β22ββ)Ο
D. 25ΟββΟ2ββ1
E. (3β22β)Ο
Solution:
Since CDβ₯AB,AC=CB and β ADB is inscribed in a semicircle, it follows that β³ABD is an isosceles right triangle, β BAD=β ABD=45β and BD=2β. Note that β EDF=β ADB=90β and DE=BEβBD=2β2β. The area of the "smile" is
