2011 AMC 12A Problems/Problem 11

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Problem

Circles $A, B,$ and $C$ each have radius 1. Circles $A$ and $B$ share one point of tangency. Circle $C$ has a point of tangency with the midpoint of $\overline{AB}.$ What is the area inside circle $C$ but outside circle $A$ and circle $B?$

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

Solution

[asy] unitsize(1.1cm); defaultpen(linewidth(.8pt)); dotfactor=4;  pair A=(0,0), B=(2,0), C=(1,-1); pair M=(1,0); pair D=(2,-1); dot (A); dot (B); dot (C); dot (D); dot (M);  draw(Circle(A,1)); draw(Circle(B,1)); draw(Circle(C,1));  draw(A--B); draw(M--D); draw(D--B);  label("$A$",A,W); label("$B$",B,E); label("$C$",C,W); label("$M$",M,NE); label("$D$",D,SE); [/asy]

The requested area is the area of $C$ minus the area shared between circles $A$, $B$ and $C$.

Let $M$ be the midpoint of $\overline{AB}$ and $D$ be the other intersection of circles $C$ and $B$.

Then area shared between $C$, $A$ and $B$ is $4$ of the regions between arc $\widehat {MD}$ and line $\overline{MD}$, which is (considering the arc on circle $B$) a quarter of the circle $B$ minus $\triangle MDB$:

$\frac{\pi r^2}{4}-\frac{bh}{2}$

$b = h = r = 1$

(We can assume this because $\angle DBM$ is 90 degrees, since $CDBM$ is a square, due the application of the tangent chord theorem at point $M$)

So the area of the small region is

$\frac{\pi}{4}-\frac{1}{2}$

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

$\pi 1^2 - 4(\frac{\pi}{4}-\frac{1}{2}) = \pi - \pi + 2 = 2$

$\boxed{\textbf{C}}$.

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)

See also

2011 AMC 12A (ProblemsAnswer KeyResources)
Preceded by
Problem 10
Followed by
Problem 12
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 12 Problems and Solutions