The area of a square is 100 square centimetres. Find the length of its diagonal in centimetres.

Introduction / Context:
This problem examines the relationship between the side and the diagonal of a square. It uses the area of the square to find the side length, and then applies the Pythagorean theorem to get the diagonal. Such questions are standard in geometry and aptitude topics involving squares and right triangles.

Given Data / Assumptions:

• Area of the square A = 100 square centimetres.
• We must find the length of the diagonal in centimetres.
• All angles of a square are right angles, and all sides are equal.

Concept / Approach:
If the side of the square is s, then its area is s^2. The diagonal of a square can be found using the Pythagorean theorem on the right triangle formed by two sides and the diagonal. In such a triangle, diagonal d satisfies d^2 = s^2 + s^2 = 2 * s^2, so d = s * sqrt(2). We first find s from the area, then find d using this relationship.

Step-by-Step Solution:
Step 1: Let side length be s. Area A = s^2. Given A = 100, so s^2 = 100. Step 2: Solve for s: s = sqrt(100) = 10 centimetres. Step 3: For a square, diagonal d = s * sqrt(2). Substitute s = 10: d = 10 * sqrt(2) centimetres.

Verification / Alternative check:
We can verify numerically by approximating sqrt(2) as 1.414. Then d is approximately 10 * 1.414 = 14.14 centimetres. If we square this approximate diagonal, we get about 200, which equals 2 * s^2, confirming that the diagonal length formula d = s * sqrt(2) is applied correctly. Since s = 10, the exact form 10 * sqrt(2) is correct.

Why Other Options Are Wrong:
Option B (10 cm): This is the side length, not the diagonal, so it is too short. Option C (20 cm): This would be twice the side length and is longer than the diagonal should be. Option D (20*sqrt(2) cm): This is even larger and corresponds to a square with side 20, not side 10. Option E (5*sqrt(2) cm): This is too small and would correspond to a square with side 5, not area 100.

Common Pitfalls:
A common error is to think that the diagonal equals side times 2 instead of side times sqrt(2). Another mistake is to miscompute the side length from the area, especially when the area is not a perfect square. In this case, the area 100 makes the side length easy, but the idea remains the same: take the square root of the area to get the side, then multiply by sqrt(2) to get the diagonal.

Final Answer:
The length of the diagonal of the square is 10*sqrt(2) cm.