1995 AHSME Problems/Problem 21

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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:

Case 1: The point "above" the given diagonal is $(4,-3)$ .

Then the point "below" the given diagonal is $(-4,3)$ .

Case 2: The point "above" the given diagonal is $(0,5)$ .

Then the point "below" the given diagonal is $(0,-5)$ .

Case 3: The point "above" the given diagonal is $(-5,0)$ .

Then the point "below" the given diagonal is $(5,0)$ .

We have only three cases since there are $8$ lattice points on the circle. $\Rightarrow \mathrm{(C)}$

1995 AHSME (Problems)
Preceded by Problem 19	Followed by Problem 21
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1995 AHSME Problems/Problem 21

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Problem

Solution

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