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Ratio of Tangent Circle Areas

Source: AMC 12 2008A Problem 13

February 17, 2008
ratiogeometryARMLAMC

Problem Statement

Points A A and B B lie on a circle centered at O O, and \angle AOB\equal{}60^\circ. A second circle is internally tangent to the first and tangent to both OA \overline{OA} and OB \overline{OB}. What is the ratio of the area of the smaller circle to that of the larger circle?
<spanclass=latexbold>(A)</span> 116<spanclass=latexbold>(B)</span> 19<spanclass=latexbold>(C)</span> 18<spanclass=latexbold>(D)</span> 16<spanclass=latexbold>(E)</span> 14 <span class='latex-bold'>(A)</span>\ \frac{1}{16} \qquad <span class='latex-bold'>(B)</span>\ \frac{1}{9} \qquad <span class='latex-bold'>(C)</span>\ \frac{1}{8} \qquad <span class='latex-bold'>(D)</span>\ \frac{1}{6} \qquad <span class='latex-bold'>(E)</span>\ \frac{1}{4}