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

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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>?
 
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>?
  
<math> \mathrm{(A) \ } 2\sqrt{2}+2\sqrt{3}\qquad \mathrm{(B) \ } 3\sqrt{3}+\sqrt{2}</math><math>\mathrm{(C) \ } 2\sqrt{6}+\sqrt{3}\qquad \mathrm{(D) \ } 3\sqrt{2}+\sqrt{6}\qquad \mathrm{(E) \ }  6\sqrt{2}-\sqrt{3}</math>
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<math> \mathrm{(A) \ } 2\sqrt{2}+2\sqrt{3}\qquad \mathrm{(B) \ } 3\sqrt{3}+\sqrt{2}</math><math>\mathrm{(C) \ } 2\sqrt{6}+\sqrt{3}\qquad \mathrm{(D) \ } 3\sqrt{2}+\sqrt{6}</math><math>\mathrm{(E) \ }  6\sqrt{2}-\sqrt{3}</math>
  
 
== Solution ==
 
== Solution ==

Revision as of 00:09, 11 July 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}$$\mathrm{(C) \ } 2\sqrt{6}+\sqrt{3}\qquad \mathrm{(D) \ } 3\sqrt{2}+\sqrt{6}$$\mathrm{(E) \ }  6\sqrt{2}-\sqrt{3}$

Solution

See also

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