Difference between revisions of "2011 AMC 12A Problems/Problem 11"
(Added solution.) |
|||
Line 63: | Line 63: | ||
<math>\boxed{\textbf{C}}</math>. | <math>\boxed{\textbf{C}}</math>. | ||
+ | |||
+ | == Solution 2 == | ||
+ | <asy> | ||
+ | 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); | ||
+ | </asy> | ||
== See also == | == See also == | ||
{{AMC12 box|year=2011|num-b=10|num-a=12|ab=A}} | {{AMC12 box|year=2011|num-b=10|num-a=12|ab=A}} |
Revision as of 21:44, 24 February 2011
Contents
[hide]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)
See also
2011 AMC 12A (Problems • Answer Key • Resources) | |
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 |