The perimeter of a rectangle is $48$ m and its area is $135$ m$^2$. Determine the sides of the rectangle.

Question

The perimeter of a rectangle is $48$ m and its area is $135$ m$^2$. Determine the sides of the rectangle.

I tried the following:

Perimeter $=48$ m
Let the length be $x$ m and the breadth be $y$ m
As we know, Perimeter $=2(x+y)=48$ $\Rightarrow x+y=24$ $[\cdots (1)]$
Area$=135$ m$^2$.
As we know, Area$=lb$
Therefore, $135$ m$^2$ $=xy$ $[\cdots (2)]$

But, how do I solve this?

Answer 1

$x+y=24$

$x\cdot y=135$

Now solve for $y$ and substitute into the other equation.

$y=24-x$

$x(24-x)=135$

$-x^2+24x-135=0$

$x^2-24x+135=0$

$(x-15)(x-9)=0$

$x=15$ or $x=9$

Plugging back into original equation you will get $x=15, y=9$ or $x=9, y=15$. They can be interchangeable because it is a rectangle. What we consider the length or the width is arbitrary because we can simply rotate the rectangle to change perspective.

Answer 2

$(x-y)^2=(x+y)^2-4xy=(24)^2-4.135=576-540=36$

Hence, $(x-y)=6$

Can you take it from here?