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Equilateral Triangles in Square

Source: 2012 AMC10B Problem #14

February 23, 2012
geometryrhombustrigonometrytrig identitiesLaw of SinesAMC

Problem Statement

Two equilateral triangles are contained in a square whose side length is 232\sqrt3. The bases of these triangles are the opposite sides of the square, and their intersection is a rhombus. What is the area of the rhombus?
<spanclass=′latex−bold′>(A)</span> 32<spanclass=′latex−bold′>(B)</span> 3<spanclass=′latex−bold′>(C)</span> 22−1<spanclass=′latex−bold′>(D)</span> 83−12<spanclass=′latex−bold′>(E)</span> 433 <span class='latex-bold'>(A)</span>\ \frac{3}{2}\qquad<span class='latex-bold'>(B)</span>\ \sqrt3\qquad<span class='latex-bold'>(C)</span>\ 2\sqrt2-1\qquad<span class='latex-bold'>(D)</span>\ 8\sqrt3-12\qquad<span class='latex-bold'>(E)</span>\ \frac{4\sqrt3}{3}
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