Question

One side of a rectangle exceeds its other side by 2 cm. If its area is 195 $c{{m}^{2}}$. Determine the sides of the rectangle.

Answer 1

Hint: We need to assume the one side of the rectangle as x and from the relation find the other side also. Using the area formula of the rectangle, we find the quadratic relation between the variable and the constant. We solve the equation to find the possible values of x. We find the two sides of the rectangle after that.

Complete step-by-step answer:
Let’s assume that one side of the rectangle is x cm. $\left( x> 0 \right)$
The other side exceeds by 2 cm. So, the other side’s measurement is $\left( x+2 \right)$ cm.
We know that if the sides of a rectangle is a and b unit then the area of that rectangle will be ab sq. unit.
So, the rectangle with sides’ measurement x and $\left( x+2 \right)$ cm, the area of that rectangle will be $x\left( x+2 \right)$ sq. cm.
It’s also given that the area of the rectangle will be 195 $c{{m}^{2}}$.
So, equating both parts we find the quadratic equation as $x\left( x+2 \right)=195$.

Now we need to solve the equation to find possible values of x.
$\begin{align}
  & x\left( x+2 \right)=195 \\
 & \Rightarrow {{x}^{2}}+2x-195=0 \\
 & \Rightarrow {{x}^{2}}+15x-13x-195=0 \\
 & \Rightarrow \left( x+15 \right)\left( x-13 \right)=0 \\
\end{align}$
Using the method of factorisation, we formed the multiple form of the quadratic equation.
We now have two possible outcomes for x which are $x=13,-15$.
Now the x is the value or length of a side. So, obviously it can’t be negative.
Possible value of x is 13 cm. The other side will be $\left( 13+2 \right)=15$ cm.
The sides of the rectangle are 13 and 15 cm.

Note: We need to always counter check the area of the rectangle from the solution. The side’s measurement was 13 and 15. So, the area will be $13\times 15=195$ sq. cm. We also need to remember to cancel out the negative answer of x as $\left( x> 0 \right)$.