Question

In a rectangle, the difference between the sum of the adjacent sides and the diagonal is half of the longer side. What is the ratio of the shorter side to the longer side?

• A. $\sqrt{3}:\sqrt{2}$
• B. $1:\sqrt{3}$
• C. $2:5$
• D. $3:4$

Answer 1

The correct option is D $3:4$

Length the adjacent sides be $l$ and $b$ and let $l$ be the longer side.

Then, diagonal $=\sqrt{l^2+b^2}$

Given, difference between the sum of the adjacent sides and the diagonal is half of the longer side.

$\Rightarrow (l+b)-\sqrt{l^2+b^2}=\frac{l}{2}$

$\Rightarrow \frac{l}{2}+b=\sqrt{l^2+b^2}$

Squaring both sides,

$\frac{l^2}{4}+b^2+lb=l^2+b^2$

$\Rightarrow \frac{l^2}{4}+lb=l^2$

$\Rightarrow lb=\frac{3l^2}{4}$

$\Rightarrow b=\frac{3l}{4}$

$\Rightarrow \frac{b}{l}=\frac{3}{4}=3:4$