2011 AMC 12A Problems/Problem 11

Problem

Circles and each have radius 1. Circles and share one point of tangency. Circle has a point of tangency with the midpoint of What is the area inside circle but outside circle and circle

$\textbf{(A)}\ 3 - \frac{\pi}{2} \qquad \textbf{(B)}\ \frac{\pi}{2} \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ \frac{3\pi}{4} \qquad \textbf{(E)}\ 1+\frac{\pi}{2}}$ (Error compiling LaTeX. Unknown error_msg)

Solution

The requested area is the area of minus the area shared between circles , and .

Let be the midpoint of and be the other intersection of circles and .

Then area shared between , and is of the regions between arc and line , which is (considering the arc on circle ) a quarter of the circle minus :

(We can assume this because is 90 degrees, since is a square, due the application of the tangent chord theorem at point )

So the area of the small region is

The requested area is area of circle minus 4 of this area:

.

Solution 2

unitsize(1.1cm);
defaultpen(linewidth(.8pt));
dotfactor=4;

pair A=(0,0), B=(2,0), C=(1,1)
pair D=(2,1);
pair E=(0,1);
pair F = (1, 2);
pair M = (1, 0);
dot (A);
dot (B);
dot (C);
dot (D);
dot (E);
dot (F);
dot (M);

draw(Circle(A,1));
draw(Circle(B,1));
draw(Circle(C,1));

draw(A--B);
draw(M--E);
draw(E--B);
draw (D--F--E--M);

label("$A$",A,W);
label("$B$",B,E);
label("$C$",C,W);
label("$M$",M,NE);
label("$D$",D,SE);
label("$E$",E,SE);
label("$F$",F,SE);
 (Error making remote request. Unknown error_msg)