Question

The diagonal of a rectangle is thrice its smaller side. Find the ratio of its sides.

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

Answer 1

The correct option is B

$2\sqrt{2}:1$

Step 1: Assuming the required measurement

Let $ABCD$ is the rectangle in which $AB=a$ is the longer side and $BC=b$ is the smaller side

According to condition of the question,

The diagonal of a rectangle is thrice its smaller side

We have diagonal $AC=3b$

Step 2: Finding the ratio

In $∆ABC$ , $\angle B=90°$ [As each angle of rectangle is of $90°$]

Therefore using Pythagoras theorem we have

$$\begin{gathered} A{C}^2=A{B}^2+B{C}^2 \\ \Rightarrow {\left(3b\right)}^2=a^2+b^2 \\ \Rightarrow 9b^2=a^2+b^2 \\ \Rightarrow 9b^2-b^2=a^2 \\ \Rightarrow 8b^2=a^2 \\ \Rightarrow a^2=8b^2 \\ \Rightarrow \frac{a^2}{b^2}=\frac{8}{1} \\ \Rightarrow \frac{a}{b}=\sqrt{\frac{8}{1}} \\ \Rightarrow \frac{a}{b}=\frac{2\sqrt{2}}{1} \\ \Rightarrow a:b=2\sqrt{2}:1 \\ \Rightarrow AB:BC=2\sqrt{2}:1 \end{gathered}$$

Hence, option (B) is the correct option.