Difference between revisions of "2007 AMC 10A Problems/Problem 15"

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Four circles of radius <math>1</math> are each tangent to two sides of a square and externally tangent to a circle of radius <math>2</math>, as shown. What is the area of the square?
 
Four circles of radius <math>1</math> are each tangent to two sides of a square and externally tangent to a circle of radius <math>2</math>, as shown. What is the area of the square?
  
{{image}}
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[[Image:2007 AMC 10A -15 for wiki.png]]
  
 
<math>\text{(A)}\ 32 \qquad \text{(B)}\ 22 + 12\sqrt {2}\qquad \text{(C)}\ 16 + 16\sqrt {3}\qquad \text{(D)}\ 48 \qquad \text{(E)}\ 36 + 16\sqrt {2}</math>
 
<math>\text{(A)}\ 32 \qquad \text{(B)}\ 22 + 12\sqrt {2}\qquad \text{(C)}\ 16 + 16\sqrt {3}\qquad \text{(D)}\ 48 \qquad \text{(E)}\ 36 + 16\sqrt {2}</math>
  
 
==Solution==
 
==Solution==
<math>22+12\sqrt{2}\ \mathrm{(B)}</math>
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The diagonal has length <math>\sqrt{2}+1+2+2+1+\sqrt{2}=6+2\sqrt{2}</math>. Therefore the sides have length <math>2+3\sqrt{2}</math>, and the area is
  
{{solution}}
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<cmath>A=(2+3\sqrt{2})^2=4+6\sqrt{2}+6\sqrt{2}+18=22+12\sqrt{2} \Rightarrow \text{(C)}</cmath>
  
 
==See Also==
 
==See Also==

Revision as of 19:01, 1 March 2008

Problem

Four circles of radius $1$ are each tangent to two sides of a square and externally tangent to a circle of radius $2$, as shown. What is the area of the square?

2007 AMC 10A -15 for wiki.png

$\text{(A)}\ 32 \qquad \text{(B)}\ 22 + 12\sqrt {2}\qquad \text{(C)}\ 16 + 16\sqrt {3}\qquad \text{(D)}\ 48 \qquad \text{(E)}\ 36 + 16\sqrt {2}$

Solution

The diagonal has length $\sqrt{2}+1+2+2+1+\sqrt{2}=6+2\sqrt{2}$. Therefore the sides have length $2+3\sqrt{2}$, and the area is

\[A=(2+3\sqrt{2})^2=4+6\sqrt{2}+6\sqrt{2}+18=22+12\sqrt{2}  \Rightarrow \text{(C)}\]

See Also

2007 AMC 10A (ProblemsAnswer KeyResources)
Preceded by
Problem 14
Followed by
Problem 16
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All AMC 10 Problems and Solutions
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