2012 AMC 10A Problem 21

Let points A = (0 ,0 ,0), B = (1, 0, 0), C = (0, 2, 0), and D = (0, 0, 3). Points E, F, G, and H are midpoints of line segments \overline{BD},\text{ } \overline{AB}, \text{ } \overline {AC}, and \overline{DC} respectively. What is the area of EFGH?

\textbf{(A)}\ \sqrt{2}\qquad\textbf{(B)}\ \frac{2\sqrt{5}}{3}\qquad\textbf{(C)}\ \frac{3\sqrt{5}}{4}\qquad\textbf{(D)}\ \sqrt{3}\qquad\textbf{(E)}\ \frac{2\sqrt{7}}{3}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.