The diagonal of a square is 10 cm long. Using properties of a square, what is the area of this square measured in square centimetres?

Introduction / Context:
This problem applies basic geometry of squares and right triangles. When the diagonal of a square is known, we can use Pythagoras theorem to relate the diagonal to the side, and then compute the area. Questions like this are common in aptitude tests to check quick reasoning with simple formulas.

Given Data / Assumptions:

• The figure is a perfect square.
• Length of the diagonal d = 10 cm.
• All angles in a square are right angles.
• We are asked to find the area in sq cm.

Concept / Approach:
For a square of side s, the diagonal d relates to the side by: d = s√2 This comes from Pythagoras theorem on a right triangle formed by two sides and the diagonal. Then area A of the square is: A = s^2 So we first find s from the diagonal, then compute s^2.

Step-by-Step Solution:
Given d = 10 cm, use d = s√2. So s = d / √2 = 10 / √2. Rationalise: 10 / √2 = (10√2) / 2 = 5√2. Now compute area: A = s^2 = (5√2)^2. (5√2)^2 = 25 * 2 = 50. Therefore, A = 50 sq cm.

Verification / Alternative check:
We can check quickly by reversing the process. If the side were 5√2 cm, the diagonal would be s√2 = 5√2 * √2 = 5 * 2 = 10 cm, which matches the given diagonal. So our derived side is consistent, and the square of that side is correctly 50.

Why Other Options Are Wrong:
100 sq cm and 200 sq cm are too large and would correspond to much bigger side lengths. 25 sq cm would correspond to side 5 cm, which would give a diagonal of 5√2 ≈ 7.07 cm, not 10 cm.

Common Pitfalls:
Students sometimes mistakenly treat the diagonal as the side and directly square 10 to get 100, ignoring the √2 relationship. Others forget to square the entire expression 5√2, or mis-handle the rationalisation. Always remember that the diagonal of a square is longer than its side by a factor of √2.

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