1995 AHSME Problems/Problem 21

Problem

Two nonadjacent vertices of a rectangle are $(4,3)$ and $(-4,-3)$ , and the coordinates of the other two vertices are integers. The number of such rectangles is

$\mathrm{(A) \ 1 } \qquad \mathrm{(B) \ 2 } \qquad \mathrm{(C) \ 3 } \qquad \mathrm{(D) \ 4 } \qquad \mathrm{(E) \ 5 }$

Solution

The distance between $(4,3)$ and $(-4,-3)$ is $\sqrt{6^2+8^2}=10$ . Therefore, if you circumscribe a circle around the rectangle, it has a center of $(0,0)$ with a radius of $10/2=5$ . There are three cases: