1995 AMC 12 第 21 题

Question

一个长方形的两个不相邻顶点为

(4,3)

和

(-4,-3)

，另外两个顶点的坐标均为整数。这样的长方形有多少个？

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

长方形的两条对角线长度相等且中点相同

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

列出从原点出发、长度为

5

的整数向量，并把互为反向的向量配对

Enumerate integer vectors of length

5

from the origin, identifying antipodal pairs

已知对角线的中点是

(0,0)

，半长为

5

。因此另一条对角线的端点必须是

(u,v)

和

(-u,-v)

，其中

u^2+v^2=25

。反过来，中点相同且长度相等的两条对角线构成一个长方形。这个圆上共有

12

个整数点：

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

一对对径点确定同一条对角线，因此有

6

种可能。其中一种就是已知对角线本身，会产生退化图形，所以还剩

6-1=5

个长方形。因此正确答案是 E。

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.

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