1995 AMC 12 Problem 21

Question

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

1

2

3

4

5

A rectangle’s diagonals have equal length and the same midpoint

Enumerate integer vectors of length

5

from the origin, identifying antipodal pairs

The given diagonal has midpoint

(0,0)

and half-length

5.

The other diagonal must therefore have endpoints

(u,v)

and

(-u,-v)

with

u^2+v^2=25.

Conversely, two equal diagonals with the same midpoint form a rectangle. There are

12

integer points on this circle:

\begin{gathered} (\pm5,0),(0,\pm5),\\ (\pm3,\pm4),(\pm4,\pm3). \end{gathered}

Antipodal points determine the same diagonal, giving

6

possibilities. One is the given diagonal itself, which is degenerate, so

6-1=5

rectangles remain. Thus the correct answer is E.

Problem 21 in other years