Problem:
Two nonadjacent vertices of a rectangle are and , and the coordinates of the other two vertices are integers. The number of such rectangles is
Answer Choices:
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Solution:
The diagonals of a rectangle are of the same length and bisect each other. The given diagonal has length and midpoint . The other diagonal must have end points on the circle of radius centered at the origin and must have integer coordinates for each end point. We must find integer solutions to . The only possible diagonals, other than the given diagonal, are the segments:
Each of these five, with the original diagonal, determines a rectangle.