# A piece of wire 44 cm long is cut into two parts and each part is bent to form a square. If the total area of the two squares is 65 sq cm, how do you find the perimeter of the two squares?

Aug 3, 2016

Here's what I got.

#### Explanation:

You know that you're working with a piece of wire that is $\text{44 cm}$ long. You then proceed to cut this piece of wire into two pieces. If you take $x$ to be the first piece, you will have

$44 - x \to$ the second piece

Now, these pieces are used to form two squares. Since a square has four equal sides, the length of one side of the first square will be $\frac{x}{4}$.

Similarly, the length of one side of the second square will be $\frac{44 - x}{4}$.

The area of a square is given by

$\textcolor{b l u e}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} {\text{area" = "side}}^{2} \textcolor{w h i t e}{\frac{a}{a}} |}}}$

In your case, the area of the first square will be

${A}_{1} = {\left(\frac{x}{4}\right)}^{2} = {x}^{2} / 16$

The are of the second square is

${A}_{2} = {\left(\frac{44 - x}{4}\right)}^{2} = {\left(44 - x\right)}^{2} / 16$

The problem tells you that the total area of the square

${A}_{\text{total}} = {A}_{1} + {A}_{2}$

is equal to ${\text{65 cm}}^{2}$, which means that you have

$65 = {x}^{2} / 16 + {\left(44 - x\right)}^{2} / 16$

This is equivalent to

${x}^{2} + {\left(44 - x\right)}^{2} = 65 \cdot 16$

${x}^{2} + {44}^{2} - 88 x + {x}^{2} = 1040$

$2 {x}^{2} - 88 x + 896 = 0$

${x}_{1 , 2} = \frac{- \left(- 88\right) \pm \sqrt{{\left(- 88\right)}^{2} - 4 \cdot 2 \cdot 896}}{2 \cdot 2}$

${x}_{1 , 2} = \frac{88 \pm \sqrt{576}}{4}$

${x}_{1 , 2} = \frac{88 \pm 24}{4} \implies \left\{\begin{matrix}{x}_{1} = \frac{88 + 24}{4} = 28 \\ {x}_{2} = \frac{88 - 24}{4} = 16\end{matrix}\right.$

Here comes the cool part. You know that the sides of the two squares are

$\text{For the 1"^("st")" square: "28/4" "color(red)("or")" "16/4" }$

$\text{For the 2"^("nd")" square: "(44-28)/4 = 16/4" " color(red)("or")" "(44-16)/4 = 28/4 " }$

As you know the perimeter of a square is given by the equation

$\textcolor{b l u e}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \text{perimeter" = 4 xx "side} \textcolor{w h i t e}{\frac{a}{a}} |}}}$

This means that the perimeters of the two squares are

$\text{For the 1"^("st") " square: "4 xx 28/4 = "28 cm " color(red)("or") " "4 xx 16/4 = "16 cm}$

$\text{For the 2"^("nd")" square: " 4 xx 16/4 = "16 cm " color(red)("or") " "4 xx 28/4 = "28 cm}$

This means that if the perimeter of the first square is $\text{28 cm}$, then the perimeter of the second square is $\text{16 cm}$.

Similarly, if if the perimeter of the first square is $\text{16 cm}$, then the perimeter of the second square is $\text{28 cm}$.