1989 AMC 12 第 3 题

Question

如图，沿两条平行于一边的直线将一个正方形切成三个矩形。若每个矩形的周长均为

24

，则原正方形的面积为

A square is cut into three rectangles along two lines parallel to a side, as shown. If the perimeter of each of the three rectangles is

24,

then the area of the original square is

24

36

64

81

96

周长相等说明三个矩形的宽相等

Equal perimeters force the three rectangle widths to be equal

用正方形边长表示一个矩形的周长

Express one rectangle’s perimeter using the square’s side length

设正方形边长为

s

。每个矩形的长均为

s

，且周长相同，所以三个宽相等，均为

\frac{s}{3}

。因此

2\left(s+\frac{s}{3}\right)=24\text{，}

故

s=9

，正方形面积为

81

。

所以正确答案是 D。

Let the square have side

s.

Since every rectangle has length

s

and the same perimeter, their three widths are equal and are each

\frac{s}{3}.

Thus

2\left(s+\frac{s}{3}\right)=24,

so

s=9

and the square’s area is

81.

Thus the correct answer is D.

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