AHSME 1979 Problem 10

If $P_1P_2P_3P_4P_5P_6$ is a regular hexagon whose apothem (distance from the center to midpoint of a side) is $2$, and $Q_i$ is the midpoint of side $P_iP_{i+1}$ for $i=1,2,3,4$, then the area of quadrilateral $Q_1Q_2Q_3Q_4$ is $\textbf{(A) }6 \textbf{(B) }2\sqrt{6} \textbf{(C) }\frac{8\sqrt{3}}{3} \textbf{(D) }3\sqrt{3} \textbf{(E) }4\sqrt{3}$