AMC 12 2002b Problem 18

A point $P$ is randomly selected from the rectangular region with vertices $(0,0),(2,0),(2,1),(0,1)$. What is the probability that $P$ is closer to the origin than it is to the point $(3,1)$?

$\mathrm{(A)}\ \frac 12 \mathrm{(B)}\ \frac 23 \mathrm{(C)}\ \frac 34 \mathrm{(D)}\ \frac 45 \mathrm{(E)}\ 1$