Difference between revisions of "2006 AMC 12A Problems/Problem 22"

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== Problem ==
 
== Problem ==
  
A circle of radius <math>r</math> is concentric with and outside a regular hexagon of side length <math>2</math>. The probability that three entire sides of hexagon are visible from a randomly chosen point on the circle is <math>1/2</math>. What is <math>r</math>?
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A [[circle]] of [[radius]] <math>r</math> is [[concentric]] with and outside a [[regular polygon | regular]] [[hexagon]] of side length <math>2</math>. The [[probability]] that three entire sides of hexagon are visible from a randomly chosen point on the circle is <math>1/2</math>. What is <math>r</math>?
  
 
<math> \mathrm{(A) \ } 2\sqrt{2}+2\sqrt{3}\qquad \mathrm{(B) \ } 3\sqrt{3}+\sqrt{2}</math><math>\rm{(C) \ } 2\sqrt{6}+\sqrt{3}\qquad \rm{(D) \ } 3\sqrt{2}+\sqrt{6}</math>
 
<math> \mathrm{(A) \ } 2\sqrt{2}+2\sqrt{3}\qquad \mathrm{(B) \ } 3\sqrt{3}+\sqrt{2}</math><math>\rm{(C) \ } 2\sqrt{6}+\sqrt{3}\qquad \rm{(D) \ } 3\sqrt{2}+\sqrt{6}</math>
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== Solution ==
 
== Solution ==
 
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{{solution}}
 
== See also ==
 
== See also ==
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* [[2006 AMC 12A Problems/Problem 21 | Previous problem]]
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* [[2006 AMC 12A Problems/Problem 23 | Next problem]]
 
* [[2006 AMC 12A Problems]]
 
* [[2006 AMC 12A Problems]]
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[[Category:Intermediate Geometry Problems]]

Revision as of 20:05, 3 November 2006

Problem

A circle of radius $r$ is concentric with and outside a regular hexagon of side length $2$. The probability that three entire sides of hexagon are visible from a randomly chosen point on the circle is $1/2$. What is $r$?

$\mathrm{(A) \ } 2\sqrt{2}+2\sqrt{3}\qquad \mathrm{(B) \ } 3\sqrt{3}+\sqrt{2}$$\rm{(C) \ } 2\sqrt{6}+\sqrt{3}\qquad \rm{(D) \ } 3\sqrt{2}+\sqrt{6}$

$\mathrm{(E) \ }  6\sqrt{2}-\sqrt{3}$

Solution

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See also

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