Regular hexagon $ABCDEF$ has side length $2$. Let $G$ be the midpoint of $\overline{AB}$, and let $H$ be the midpoint of $\overline{DE}$. What is the perimeter of $GCHF$? $\textbf{(A) }\ 4\sqrt3 \qquad$ $ \textbf{(B) }\ 8 \qquad$ $ \textbf{(C) }\ 4\sqrt5 \qquad$ $ \textbf{(D) }\ 4\sqrt7 \qquad$ $ \textbf{(E) }\ 12$
Solution
A regular hexagon of side $2$ has circumradius $2$. Using a flat-top orientation, take $A=(0,2)$, $B=(\sqrt3,1)$, $C=(\sqrt3,-1)$, $D=(0,-2)$, $E=(-\sqrt3,-1)$, and $F=(-\sqrt3,1)$. Then $G=(\sqrt3/2,3/2)$ and $H=(-\sqrt3/2,-3/2)$.
By the distance formula, $$GC=\sqrt{(\sqrt3/2)^2+(5/2)^2}=\sqrt7,\qquad CH=\sqrt{(3\sqrt3/2)^2+(1/2)^2}=\sqrt7.$$ By symmetry, all four sides of $GCHF$ have length $\sqrt7$.
Its perimeter is $4\sqrt7$, so the answer is $\textbf{(D)}$.
Regular hexagon $ABCDEF$ has side length $2$. Let $G$ be the midpoint of $\overline{AF}$, and let $H$ be the midpoint of $\overline{CD}$. What is the perimeter of $GCHF$?
$\textbf{(A) }\ 4\sqrt3 \qquad$ $ \textbf{(B) }\ 8 \qquad$ $ \textbf{(C) }\ 4\sqrt5 \qquad$ $ \textbf{(D) }\ 4\sqrt7 \qquad$ $ \textbf{(E) }\ 12$ 
