What is the area (in square centimetres) of a rectangle whose diagonal is 25 cm and one of its sides is 24 cm long?

Introduction / Context:
This problem combines the Pythagorean theorem with the area formula for a rectangle. A rectangle has opposite sides equal and all angles equal to 90°, so its diagonal forms a right triangle with the two adjacent sides as legs. By using the diagonal and one side, we can find the other side using Pythagoras, and then compute the area as length multiplied by breadth.

Given Data / Assumptions:

• The shape is a rectangle.
• The length of the diagonal is 25 cm.
• One side of the rectangle is 24 cm.
• We must find the area of the rectangle in square centimetres.

Concept / Approach:
Let the sides of the rectangle be a and b, and the diagonal be d. The Pythagorean theorem gives:
d^2 = a^2 + b^2 Once both side lengths are known, the area is given by:
Area = a * b We are given d and one side, so we solve for the other side and then compute the area.

Step-by-Step Solution:
Step 1: Let one side of the rectangle be a = 24 cm and the other side be b (unknown). Step 2: The diagonal is given as d = 25 cm. Step 3: Apply the Pythagorean theorem: d^2 = a^2 + b^2. Step 4: Substitute the values: 25^2 = 24^2 + b^2. Step 5: Compute squares: 625 = 576 + b^2. Step 6: Rearrange to find b^2: b^2 = 625 - 576 = 49. Step 7: Take the square root: b = sqrt(49) = 7 cm. Step 8: Compute the area: Area = a * b = 24 * 7. Step 9: Multiply: 24 * 7 = 168 sq cm.

Verification / Alternative check:
We can check the diagonal using the side lengths 24 cm and 7 cm. The diagonal squared should equal 24^2 + 7^2 = 576 + 49 = 625, so the diagonal is sqrt(625) = 25 cm, which matches the given value. The area is then clearly 24 * 7 = 168 sq cm, confirming that the computed area is consistent with both the Pythagorean theorem and the given dimensions.

Why Other Options Are Wrong:

• 186, 144, 132, and 150: These values come from incorrect combinations or arithmetic errors. None of them correspond to 24 multiplied by the correctly derived second side of 7 cm.

Common Pitfalls:
Some students mistakenly add or subtract the diagonal and given side instead of using the Pythagorean theorem. Others might use the wrong formula for area. Remember that a rectangle's diagonal, together with its sides, always satisfies a^2 + b^2 = d^2, and area is simply the product of the two side lengths once they are known.

Final Answer:
Therefore, the area of the rectangle is 168 sq cm.