Question

If the perimeter of a rectangle is p and its diagonal is d, the difference between the length and width of the rectangle is:

• A. $\frac{\sqrt{8d^2-p^2}}{2}$
• B. $\frac{\sqrt{8d^2+p^2}}{2}$
• C. $\frac{\sqrt{6d^2-p^2}}{2}$
• D. $\frac{\sqrt{6d^2+p^2}}{2}$
• E. $\frac{\sqrt{8d^2-p^2}}{4}$

Answer 1

The correct option is A $\frac{\sqrt{8d^2-p^2}}{2}$

Perimeter of the rectangle = P

$2(l+b)=P⟹l+b=\frac{P}{2}⟶\left(1\right)$

diagonal of the rectangle = d

$\sqrt{l^2+b^2}=d⟹l^2+b^2=d^2{\left(1\right)}^2⟹d^2+2lb=\frac{P^2}{4}⟹2lb=\frac{P^2-4d^2}{4}⟹l^2+b^2-2lb=d^2-\frac{P^2-4d^2}{4}⟹{(l-b)}^2=\frac{{8d^2-P}^2}{4}⟹(l-b)=\frac{\sqrt{{8d^2-P}^2}}{2}$

$\therefore$ Difference between length and breadth = $\frac{\sqrt{{8d^2-P}^2}}{2}$