1992 AHSME Problems/Problem 26

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Problem

$[asy] fill((1,0)--arc((1,0),2,180,225)--cycle,grey); fill((-1,0)--arc((-1,0),2,315,360)--cycle,grey); fill((0,-1)--arc((0,-1),2-sqrt(2),225,315)--cycle,grey); fill((0,0)--arc((0,0),1,180,360)--cycle,white); draw((1,0)--arc((1,0),2,180,225)--(1,0),black+linewidth(1)); draw((-1,0)--arc((-1,0),2,315,360)--(-1,0),black+linewidth(1)); draw((0,0)--arc((0,0),1,180,360)--(0,0),black+linewidth(1)); draw(arc((0,-1),2-sqrt(2),225,315),black+linewidth(1)); draw((0,0)--(0,-1),black+linewidth(1)); MP("C",(0,0),N);MP("A",(-1,0),N);MP("B",(1,0),N); MP("D",(0,-.8),NW);MP("E",(1-sqrt(2),-sqrt(2)),SW);MP("F",(-1+sqrt(2),-sqrt(2)),SE); [/asy]$

Semicircle $AB$ has center $C$ and radius $1$. Point $D$ is on $AB$ and $\overline{CD}\orthogonal\overline{AB}$ (Error compiling LaTeX. ! Undefined control sequence.). Extend $\overline{BD}$ and $\overline{AD}$ to $E$ and $F$, respectively, so that circular arcs $AE$ and $BF$ have $B$ and $A$ as their respective centers. Circular arc $EF$ has center $D$. The area of the shaded "smile" $AEFBDA$, is

$\text{(A) } \quad \text{(B) } \quad \text{(C) } \quad \text{(D) } \quad \text{(E) }$

Solution

$\fbox{B}$