A rectangle has a diagonal of length 51 cm and one of its sides is 24 cm. What is the area of this rectangle in square centimetres?

Introduction / Context:
This question involves a rectangle where one side and the diagonal are known. By using the Pythagoras theorem on the right triangle formed by the two sides and the diagonal, we can find the length of the missing side. Once both side lengths are determined, the area of the rectangle is straightforward to calculate as the product of its length and breadth. This type of problem checks understanding of right triangles and basic mensuration.

Given Data / Assumptions:

• The figure is a rectangle with right angles at each corner.
• The diagonal of the rectangle measures 51 cm.
• One side (breadth or length) measures 24 cm.
• We need to find the area in square centimetres.
• Standard Pythagoras theorem applies: diagonal^2 = side1^2 + side2^2.

Concept / Approach:
In a rectangle, the diagonal acts as the hypotenuse of a right triangle formed by the length and breadth. If we let the unknown side be x, then 51^2 = 24^2 + x^2. Solving this equation gives the value of x. After finding x, the area A of the rectangle is simply A = 24 * x. Performing these steps in the correct order ensures an accurate and efficient solution.

Step-by-Step Solution:
Let the known side be 24 cm and the unknown side be x cm.Using Pythagoras theorem: diagonal^2 = side1^2 + side2^2.So 51^2 = 24^2 + x^2.Compute squares: 51^2 = 2601 and 24^2 = 576, so 2601 = 576 + x^2.Thus x^2 = 2601 − 576 = 2025, so x = √2025 = 45 cm.Area A = 24 * 45 = 1080 sq cm.

Verification / Alternative check:
We can quickly check that 24, 45, and 51 form a Pythagorean triple. Calculating 24^2 + 45^2 gives 576 + 2025 = 2601, which is indeed 51^2. This confirms that the triangle is right angled and that the sides are consistent with the diagonal. Multiplying 24 by 45 again gives 1080 sq cm, so there is no arithmetic mistake in the area computation.

Why Other Options Are Wrong:
Option 540 sq cm is exactly half of the correct area and might result from multiplying 24 by 22.5 or from some misread. Option 810 sq cm results from mixing up numbers incorrectly such as using 30 instead of 45. Option 360 sq cm is far too small and would correspond to an unknown side of only 15 cm, which is inconsistent with Pythagoras theorem. Option 720 sq cm also does not satisfy the length needed to match the diagonal of 51 cm.

Common Pitfalls:
Common mistakes include squaring the diagonal incorrectly, forgetting to subtract 24^2 from 51^2, or taking the square root of an incorrect intermediate result. Some students also confuse which side is given and incorrectly write 24^2 = 51^2 + x^2, which reverses the roles of hypotenuse and legs. Keeping diagonal as the largest side in the Pythagoras equation and working stepwise helps avoid such issues.

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