If the diagonal of a rectangle is 3 times of its smallest side, then ratio of its sides is

Appeared in: UPPSC 2017

Explanation

The problem is solved using the Pythagorean theorem, which relates the sides of a rectangle to its diagonal: (length)² + (breadth)² = (diagonal)².
Let the smaller side (breadth) be 'b'. The problem states the diagonal is 3 times the smallest side, so the diagonal is '3b'.
Substituting into the theorem: (length)² + b² = (3b)², which simplifies to (length)² = 9b² - b² = 8b².
Taking the square root, the length is √8 * b, which simplifies to 2√2 * b.
The ratio of the sides (length : breadth) is therefore (2√2 * b) : b, which simplifies to 2√2 : 1.

Option A: This ratio would result if the length 'l' were √3 times the breadth 'b'. In that case, l² = 3b², and the diagonal squared would be d² = (√3b)² + b² = 3b² + b² = 4b². This would mean the diagonal is 2b, not 3b as stated in the problem.
Option B: This ratio implies the length 'l' is 2√3 times the breadth 'b'. This would mean l² = (2√3b)² = 12b². The diagonal squared would then be d² = 12b² + b² = 13b², making the diagonal √13 * b, which contradicts the given information.
Option D: This ratio would result if the length 'l' were √2 times the breadth 'b'. This would mean l² = 2b², and the diagonal squared would be d² = 2b² + b² = 3b². This would make the diagonal √3 * b, not 3b.

Clinical Relevance

Nursing practice connection: This is primarily an exam-oriented knowledge point with limited direct bedside application, so retain Application of the Pythagorean theorem to find the ratio of the sides of a rectangle as background academic context rather than a clinical decision trigger.
This question tests foundational mathematical and logical reasoning skills, which are part of the general aptitude section of many nursing entrance exams.
Strong analytical skills are essential for nurses for tasks like medication dosage calculation, interpreting lab results, and managing fluid balances.
What if? If the diagonal was 2 times the smallest side, the equation would be l² + b² = (2b)², leading to l² = 3b² and a ratio of √3 : 1.

How to Approach the Question

First, identify the geometric shape (a rectangle) and the components mentioned (sides and diagonal).
Translate the word problem into a mathematical equation. Let the smaller side be 'b' and the larger side be 'l'. The diagonal 'd' is given as '3b'.
Recall the relevant geometric formula. For a rectangle, the sides and diagonal form a right-angled triangle, so apply the Pythagorean theorem: l² + b² = d².
Substitute the known relationship (d = 3b) into the formula: l² + b² = (3b)².
Solve the algebraic equation for 'l' in terms of 'b'.
Finally, express the ratio of the sides (l : b) and simplify it to match one of the options.

Concept Tested & Keywords

Concept Tested: Application of the Pythagorean theorem to find the ratio of the sides of a rectangle.
Stem keywords: diagonal of a rectangle, smallest side, ratio of its sides
Lead-in keywords: ratio

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