AHSME 1996 Problem 11

Given a circle of radius $2$, there are many line segments of length $2$ that are tangent to the circle at their midpoints. Find the area of the region consisting of all such line segments. $\text{(A)}\ \frac{\pi} 4 \text{(B)}\ 4-\pi \text{(C)}\ \frac{\pi} 2 \text{(D)}\ \pi \text{(E)}\ 2\pi$