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A Circle and an Equilateral Triangle

Source: 2010 AMC 10B Problem 19

February 25, 2010
geometrytrigonometryquadraticsPythagorean Theoremtrig identitiesLaw of Cosinesperpendicular bisector

Problem Statement

A circle with center O O has area 156π 156\pi. Triangle ABC ABC is equilateral, BC‾ \overline{BC} is a chord on the circle, OA \equal{} 4\sqrt3, and point O O is outside △ABC \triangle ABC. What is the side length of △ABC \triangle ABC? <spanclass=′latex−bold′>(A)</span> 23<spanclass=′latex−bold′>(B)</span> 6<spanclass=′latex−bold′>(C)</span> 43<spanclass=′latex−bold′>(D)</span> 12<spanclass=′latex−bold′>(E)</span> 18 <span class='latex-bold'>(A)</span>\ 2\sqrt3 \qquad<span class='latex-bold'>(B)</span>\ 6 \qquad<span class='latex-bold'>(C)</span>\ 4\sqrt3 \qquad<span class='latex-bold'>(D)</span>\ 12 \qquad<span class='latex-bold'>(E)</span>\ 18
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