The diagonal of a square measures 10 cm. What is the area of this square in square centimetres?

Introduction / Context:
This question tests the relation between the side and the diagonal of a square and then uses that relation to compute the area. In a square, all angles are right angles, and the diagonal splits the square into two congruent right triangles. Recognizing the 45 degree angles and the Pythagoras relation makes this a quick formula based problem suitable for aptitude exams.

Given Data / Assumptions:

• The figure is a square, so all sides are equal and all angles are right angles.
• The diagonal of the square measures 10 cm.
• We are asked to find the area of the square in square centimetres.
• Standard right triangle relationships apply.
• No rounding is necessary because the numbers are simple.

Concept / Approach:
If the side of a square is s, then the diagonal d is given by d = s√2. This comes from applying Pythagoras theorem to the right triangle formed by two sides and the diagonal. Once we know the diagonal, we can solve for the side using s = d / √2. The area A of the square is then s^2. There is also a direct formula A = d^2 / 2, which is derived by substituting s = d / √2 into s^2.

Step-by-Step Solution:
Given diagonal d = 10 cm.Use the direct formula for area of a square in terms of diagonal: A = d^2 / 2.Compute d^2 = 10^2 = 100.So A = 100 / 2 = 50 sq cm.Therefore, the area of the square is 50 square centimetres.

Verification / Alternative check:
Alternatively, compute the side length first. From d = s√2, we have s = d / √2 = 10 / √2. Rationalising the denominator gives s = (10√2) / 2 = 5√2. The area A = s^2 = (5√2)^2 = 25 * 2 = 50 sq cm. This method uses explicit side length but arrives at the same area, confirming the correctness of the result.

Why Other Options Are Wrong:
Option 100 sq cm corresponds to taking area as d^2 instead of d^2 / 2, which is incorrect. Option 200 sq cm is even larger and may come from using 2 * d^2. Option 25 sq cm is half of the correct answer and might come from dividing incorrectly. Option 75 sq cm is an arbitrary intermediate that does not follow from any valid formula relating side and diagonal of a square.

Common Pitfalls:
Students sometimes confuse the relationship between side and diagonal for squares and rectangles. Another typical error is to forget that Pythagoras theorem applies and assume the diagonal equals the sum of sides instead of the root of the sum of squares. Using the correct formula A = d^2 / 2 for squares resolves these mistakes quickly.

Final Answer:
The area of the square is 50 sq cm.