AMC 8 1989 Problem 24

AMC8 1989 年第 24 题

四边形与多边形★★★☆☆

Suppose a square piece of paper is folded in half vertically. The folded paper is then cut in half along the dashed line. Three rectangles are formed-a large one and two small ones. What is the ratio of the perimeter of one of the small rectangles to the perimeter of the large rectangle?

假设一张正方形纸被垂直对折。折叠后的纸再沿虚线切成两半。形成三个矩形——一个大矩形和两个小矩形。一个小矩形的周长与大矩形的周长之比是多少？

A. $\displaystyle \frac{1}{2}$
B. $\displaystyle \frac{2}{3}$
C. $\displaystyle \frac{3}{4}$
D. $\displaystyle \frac{4}{5}$
E. $\displaystyle \frac{5}{6}$

正确答案 · Correct Answer

解析 · Solution

Solution 1

In this diagram, the dashed segment is the fold while the blue segments are cuts.

Let

s

be the side length of the square. After being cut, the paper will be separated into two

\displaystyle s \times \frac{s}{4}

rectangles and a

\displaystyle s \times \frac{s}{2}

rectangle. The larger rectangle has perimeter

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

while the smaller rectangle has perimeter

\displaystyle 2\left(s + \frac{s}{4}\right) = \frac{5s}{2}

. Therefore, the ratio of the perimeters is

\displaystyle \left( \frac{5s}{2} \right) \div (3s) = \frac{5}{6}

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