2012 AMC 12B Problems/Problem 21

Problem

Square $AXYZ$ is inscribed in equiangular hexagon $ABCDEF$ with $X$ on $\overline{BC}$ , $Y$ on $\overline{DE}$ , and $Z$ on $\overline{EF}$ . Suppose that $AB=40$ , and $EF=41(\sqrt{3}-1)$ . What is the side-length of the square?

$\textbf{(A)}\ 29\sqrt{3} \qquad\textbf{(B)}\ \frac{21}{2}\sqrt{2}+\frac{41}{2}\sqrt{3}\qquad\textbf{(C)}\ 20\sqrt{3}+16$

$\textbf{(D)}\ 20\sqrt{2}+13\sqrt{3} \qquad\textbf{(E)}\ 21\sqrt{6}$

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2012 AMC 12B Problems/Problem 21

Problem