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Difference between revisions of "2007 AMC 10A Problems/Problem 24"

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Circles centered at <math>A</math> and <math>B</math> each have radius <math>2</math>, as shown. Point <math>O</math> is the midpoint of <math>\overline{AB}</math>, and <math>OA = 2\sqrt {2}</math>. Segments <math>OC</math> and <math>OD</math> are tangent to the circles centered at <math>A</math> and <math>B</math>, respectively, and <math>EF</math> is a common tangent. What is the area of the shaded region <math>ECODF</math>?
 
Circles centered at <math>A</math> and <math>B</math> each have radius <math>2</math>, as shown. Point <math>O</math> is the midpoint of <math>\overline{AB}</math>, and <math>OA = 2\sqrt {2}</math>. Segments <math>OC</math> and <math>OD</math> are tangent to the circles centered at <math>A</math> and <math>B</math>, respectively, and <math>EF</math> is a common tangent. What is the area of the shaded region <math>ECODF</math>?
  
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[[Image:2007 AMC 10A problem 24.png]]
  
 
<math>\text{(A)}\ \frac {8\sqrt {2}}{3} \qquad \text{(B)}\ 8\sqrt {2} - 4 - \pi \qquad \text{(C)}\ 4\sqrt {2} \qquad \text{(D)}\ 4\sqrt {2} + \frac {\pi}{8} \qquad \text{(E)}\ 8\sqrt {2} - 2 - \frac {\pi}{2}</math>
 
<math>\text{(A)}\ \frac {8\sqrt {2}}{3} \qquad \text{(B)}\ 8\sqrt {2} - 4 - \pi \qquad \text{(C)}\ 4\sqrt {2} \qquad \text{(D)}\ 4\sqrt {2} + \frac {\pi}{8} \qquad \text{(E)}\ 8\sqrt {2} - 2 - \frac {\pi}{2}</math>
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==Solution==
 
==Solution==
 
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We can't have a solution if we don't know the image, which can be found [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=751401#751401 here.]
 
  
 
==See also==
 
==See also==

Revision as of 10:33, 15 April 2008

Problem

Circles centered at $A$ and $B$ each have radius $2$, as shown. Point $O$ is the midpoint of $\overline{AB}$, and $OA = 2\sqrt {2}$. Segments $OC$ and $OD$ are tangent to the circles centered at $A$ and $B$, respectively, and $EF$ is a common tangent. What is the area of the shaded region $ECODF$?

2007 AMC 10A problem 24.png

$\text{(A)}\ \frac {8\sqrt {2}}{3} \qquad \text{(B)}\ 8\sqrt {2} - 4 - \pi \qquad \text{(C)}\ 4\sqrt {2} \qquad \text{(D)}\ 4\sqrt {2} + \frac {\pi}{8} \qquad \text{(E)}\ 8\sqrt {2} - 2 - \frac {\pi}{2}$

Solution

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See also

2007 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 23 		Followed by
Problem 25
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 10 Problems and Solutions
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