The diagonal of a rectangle is thrice its smaller side. The ratio of the length to the breadth of the rectangle is

### Concept & Ratio Mapping This problem uses the Pythagorean theorem to find the ratio between the sides without needing any absolute measurements. By setting the smaller side as a base unit, the longer side can be expressed purely in terms of that unit. ### Step-by-Step Solution 1. Let the smaller side (breadth) of the rectangle be $b$. 2. The diagonal ($d$) is given as thrice the smaller side, so $d = 3b$. 3. Let the larger side (length) be $l$. According to the Pythagorean theorem: $$ d^2 = l^2 + b^2 $$ 4. Substitute the known expressions into the theorem: $$ (3b)^2 = l^2 + b^2 $$ $$ 9b^2 = l^2 + b^2 $$ 5. Isolate $l^2$ to express length in terms of breadth: $$ l^2 = 9b^2 - b^2 $$ $$ l^2 = 8b^2 $$ 6. Take the square root of both sides to find $l$: $$ l = \sqrt{8b^2} = \sqrt{4 \times 2} \cdot b = 2\sqrt{2}b $$ 7. Find the required ratio of length to breadth ($l : b$): $$ \text{Ratio} = \frac{l}{b} = \frac{2\sqrt{2}b}{b} = \frac{2\sqrt{2}}{1} $$ $$ \text{Ratio} = 2\sqrt{2} : 1 $$ ### Exam Strategy & Shortcut Let the smaller side be exactly $1$. The diagonal is then $3$. Apply Pythagorean theorem mentally: $3^2 - 1^2 = 9 - 1 = 8$. The missing side is $\sqrt{8}$, which simplifies to $2\sqrt{2}$. The ratio is simply the sides compared: $2\sqrt{2} : 1$. ### Common Pitfall Be careful to identify which ratio the question asks for. It asks for "length to breadth", meaning the larger side first. Reversing this to $1 : 2\sqrt{2}$ would be incorrect. ### Final Answer Therefore, the correct answer is **$2\sqrt{2} : 1$**.