A rectangle has a diagonal of length 25 cm and one of its sides has length 24 cm. Using the Pythagorean theorem, what is the area (in cm²) of the rectangle?

Introduction:
Every rectangle can be viewed as made up of right triangles when a diagonal is drawn. The diagonal acts as the hypotenuse of a right triangle whose legs are the sides of the rectangle. This allows us to apply the Pythagorean theorem to find the missing side and then compute the area.

Given Data / Assumptions:

• Rectangle with diagonal d = 25 cm.
• One side (say length) is 24 cm.
• We must find the area, which is length * breadth.

Concept / Approach:
For a rectangle with sides a and b and diagonal d, the relationship by the Pythagorean theorem is: d² = a² + b². Once the unknown side is found, area = a * b. Here, we know d and one side, so we solve for the other side and multiply.

Step-by-Step Solution:
Let the given side be a = 24 cm, the unknown side be b, and diagonal d = 25 cm. Apply Pythagorean theorem: d² = a² + b². So 25² = 24² + b². Compute 25² = 625 and 24² = 576. Thus 625 = 576 + b². So b² = 625 − 576 = 49. Therefore b = √49 = 7 cm. Area = a * b = 24 * 7 = 168 cm².

Verification / Alternative check:
The triple (7, 24, 25) is a known Pythagorean triple. This confirms that a rectangle with sides 7 and 24 has diagonal 25. The area then is 7 * 24 = 168 cm², consistent with the calculation.

Why Other Options Are Wrong:
Options 132, 144, and 186 correspond to either incorrect side lengths or arbitrary multiplications that do not satisfy the Pythagorean relationship with a 25 cm diagonal. Only 168 arises from the correct side pair (24, 7) that forms a 7–24–25 right triangle.

Common Pitfalls:
Students sometimes incorrectly take the diagonal as the sum of the sides or misapply the theorem by adding the diagonal to a side. Always remember that for right triangles, the hypotenuse squared equals the sum of the squares of the legs, not the sum of the legs themselves.

Final Answer:
The area of the rectangle is 168 cm².