# Problem of the Week Problem E and Solution A Rectangle and a Square

## Problem

Simeon has a rope that is $$108$$ cm long and is asked to cut the rope once so that one of the pieces can be arranged, with its two ends touching, to form a square, and the other piece can be arranged, with its two ends touching, to form a rectangle with one side length of $$6$$ cm. Furthermore, the area of the square will be equal to the area of the rectangle.

Where should Simeon make the cut to the original piece of rope?

## Solution

Let the length of the piece of rope used to form the square be $$4x$$ cm. This is also equal to the perimeter of the square. Then the side length of the square is $$4x\div 4=x$$ cm. The area of the square is $x \times x=x^2\text{ cm}^2\tag{1}$

The length of the piece of rope used to form the rectangle is $$(108-4x)$$ cm. This is also equal to the perimeter of the rectangle. If one side length of the rectangle is 6 cm, then there is $$108-4x-6-6=(96-4x)$$ cm left to form the lengths of the two other sides of the rectangle. Therefore, the other side length of the rectangle is $$\frac{96-4x}{2}=(48-2x)$$ cm. Thus, the area of the rectangle is $(6)(48-2x)=(288-12x)\text{ cm}^2\tag{2}$

We are given that the area of the square is equal to the area of the rectangle. So, by equating equations $$(1)$$ and $$(2)$$, we obtain \begin{aligned} x^2&=288-12x\\ x^2+12x-228&=0\\ (x-12)(x+24)&=0 \end{aligned} Thus, $$x=12$$ or $$x=-24$$. Since $$x$$ is the length of the side of the square, we must have $$x>0$$. Therefore, $$x=12$$ cm. Then the length of rope used to form the square is $$4x=4(12)=48$$ cm.

Therefore, the cut should be made $$48$$ cm from one end (and so $$60$$ cm from the other end), creating a $$60$$ cm piece for the rectangle and a $$48$$ cm piece for the square.

Note:
The area of the square is $$12\times 12=144\text{ cm}^2$$.
The length of the other side of the rectangle is $$48-2x=48-24=24$$ cm. The area of the rectangle is $$24\times 6=144\text{ cm}^2$$.
(These calculations were not required but are provided as a check of the correctness of the result.)