2009 Cayley Problem 24

The field shown has been planted uniformly with wheat. At harvest, the wheat at any point in the field is brought to the nearest point on the field's perimeter. The fraction of the crop that is brought to the longest side is

\textbf{(A)}\ \frac{1}{3}\quad \textbf{(B)}\ \frac{5}{12}\quad \textbf{(C)}\ \frac{1}{2}\quad \textbf{(D)}\ \frac{2}{5}\quad \textbf{(E)}\ \frac{4}{9}

If there are no answer choices shown, enter a numerical answer.

Full credit to this problem is given to the CEMC, you may view all Cayley contests here.