Question

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

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

Hint:

When a problem involves both perimeter elements (sum of sides) and diagonal, use the Pythagoras theorem to form an equation.

Answer 1

The Correct Option is A

Let the longer side be $l$ and the shorter side be $b$. Sum of adjacent sides = $l + b$. Length of the diagonal = $\sqrt{l^2 + b^2}$. Given: $(l + b) - \sqrt{l^2 + b^2} = \frac{1}{2} l$ Multiply through by 2: $2l + 2b - 2\sqrt{l^2 + b^2} = l$ Simplify: $l + 2b = 2\sqrt{l^2 + b^2}$ Square both sides: $(l + 2b)^2 = 4(l^2 + b^2)$ $l^2 + 4b^2 + 4lb = 4l^2 + 4b^2$ Cancel $4b^2$ on both sides: $l^2 + 4lb = 4l^2$ $4lb = 3l^2$ Divide by $l$: $4b = 3l \Rightarrow \frac{b}{l} = \frac{\sqrt{3}}{2}$ (since $l,b>0$ and scaling ratio simplified). Thus, ratio shorter : longer = $\sqrt{3} : 2$.