The diagonal of a square is 'a' cm, then the area of the square is :

Question

Accepted Answer

To determine the area of a square when the diagonal is given as 'a' cm, we need to use the relationship between the side of the square and its diagonal.

### Explanation for Option 4:

1. **Relationship between diagonal and side of a square:**
   - In a square, the diagonal divides the square into two right-angled triangles. If 's' is the side of the square, then by the Pythagorean theorem, the diagonal 'd' is given by:
     $$
     d = \sqrt{s^2 + s^2} = \sqrt{2s^2} = s\sqrt{2}
     $$
   - Given that the diagonal is 'a', we have:
     $$
     a = s\sqrt{2}
     $$
   - Solving for 's', we get:
     $$
     s = \frac{a}{\sqrt{2}}
     $$

2. **Calculating the area of the square:**
   - The area of a square is given by the square of its side:
     $$
     \text{Area} = s^2 = \left(\frac{a}{\sqrt{2}}\right)^2 = \frac{a^2}{2}
     $$
   - Therefore, the area of the square is $\frac{a^2}{2} \ \text{cm}^2$, which corresponds to Option 4.

### Explanation for why other options are incorrect:

1. **Option 1: $2a \ \text{cm}^2$**
   - This option incorrectly suggests that the area is directly proportional to the diagonal without considering the square root relationship. The area should be derived from the square of the side, not the diagonal itself.

2. **Option 2: $\frac{a}{\sqrt{2}} \ \text{cm}^2$**
   - This option represents the side length of the square, not the area. The area is the square of the side length, not the side length itself.

3. **Option 3: $\frac{a^2}{4} \ \text{cm}^2$**
   - This option does not correctly apply the relationship between the diagonal and the side of the square. The correct formula for the area involves dividing by 2, not 4.

In conclusion, Option 4 is the correct answer because it accurately reflects the mathematical relationship between the diagonal and the area of a square.