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Folded rectangle

时间:2025-08-20 17:36来源: 作者:admin 点击: 4 次
Problem Student Solutions Problem A rectangular piece of paper, 3 cm by 4 cm, is folded along its diagonal, as shown below.  Im

Problem Student Solutions

Problem

A rectangular piece of paper, 3 cm by 4 cm, is folded along its diagonal, as shown below.

 

Image

Folded rectangle



What is the perimeter of the pentagon formed (shown below in blue)?

Image

Folded rectangle



This problem is adapted from the World Mathematics Championships

 

Student Solutions

By Pythagoras' Theorem, the diagonal side on the yellow triangle below is $\sqrt{3^2+4^2}=5$ cm.

 

Image

Folded rectangle



In the diagram below, the red triangles are similar. The length marked $x$ is required to find the perimeter of the pentagon.

 

Image

Folded rectangle



By Pythagoras' Theorem, $(4-x)^2=3^2+x^2$.

$$\begin{align}&(4-x)^2=3^2+x^2\\

&\Rightarrow16-8x+x^2=9+x^2\\

&\Rightarrow7=8x\\

&\Rightarrow x=\frac{7}{8}\end{align}$$

So the perimeter of the pentagon is $5+3+\dfrac{7}{8}+\dfrac{7}{8}+3=12\dfrac{3}{4}$ cm.

 

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