1983 AMC 12 第 24 题

Question

周长（以厘米计）与面积（以

\text{cm}^2

计）的数值相等的互不全等直角三角形有多少个？

How many non-congruent right triangles are there such that the perimeter in cm and area in

\text{cm}^2

are numerically equal?

一个也没有

none

1

2

4

无穷多个

infinitely many

从任意形状的直角三角形开始，将所有边长缩放

k

倍

Start with any shape of right triangle and scale all its side lengths by

k

缩放后，周长乘以

k

，而面积乘以

k^2

Under scaling, perimeter is multiplied by

k

while area is multiplied by

k^2

从任意一个周长为

P

、面积为

K

的直角三角形开始。将每条边缩放

k=\frac{P}{K}

倍后，周长为

kP

，面积为

k^2K

，两者相等。互不相似的直角三角形形状有无穷多个，所以所得三角形也互不全等。

所以正确答案是 E。

Start with any right triangle having perimeter

P

and area

K.

Scaling every side by

k=\frac{P}{K}

produces perimeter

kP

and area

k^2K,

which are equal. Infinitely many nonsimilar right-triangle shapes exist, and the resulting triangles are therefore noncongruent.

Therefore, the correct answer is E.

其他年份的第 24 题