A rectangle has a diagonal of length 51 cm and one of its sides is 24 cm. Using the Pythagoras relation, what is the area of this rectangle in square centimetres?

Introduction / Context:
This problem connects coordinate geometry ideas with mensuration of a rectangle. Knowing the diagonal and one side, we can find the other side using Pythagoras theorem, and then compute the area. It is a common pattern in aptitude tests to indirectly give the second side in this way.

Given Data / Assumptions:

• Shape is a rectangle, so all angles are 90°.
• Diagonal length d = 51 cm.
• One side (take it as length) L = 24 cm.
• Other side (breadth) is unknown, denote it by B.
• We need the area A = L * B in sq cm.

Concept / Approach:
In a rectangle, the diagonal, length, and breadth form a right triangle. By Pythagoras theorem: d^2 = L^2 + B^2 We can solve this equation for B, then multiply by L to obtain the area: A = L * B

Step-by-Step Solution:
Given d = 51 cm and L = 24 cm. Use d^2 = L^2 + B^2. Compute squares: 51^2 = 2601 and 24^2 = 576. So 2601 = 576 + B^2. Therefore, B^2 = 2601 - 576 = 2025. Take square root: B = √2025 = 45 cm. Now area A = L * B = 24 * 45. 24 * 45 = 1080 sq cm.

Verification / Alternative check:
We can check quickly: 24^2 + 45^2 = 576 + 2025 = 2601, and √2601 = 51, which matches the given diagonal. So the side lengths 24 cm and 45 cm are consistent, and their product gives the correct area.

Why Other Options Are Wrong:
540 and 810 are too small and arise if you halve one side or miscompute the breadth. 360 corresponds to 24 × 15, which would give a much shorter diagonal and does not satisfy Pythagoras theorem with 51 cm.

Common Pitfalls:
Errors often occur when squaring or subtracting large numbers, or when taking the square root. Some students mistakenly use perimeter formulas instead of applying Pythagoras theorem. Always remember that diagonal, length, and breadth form a right triangle in a rectangle.

Final Answer:
The area of the rectangle is 1080 sq cm.