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

Question

Accepted Answer

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

1. **Understanding the Relationship:**
   - In a square, the diagonal divides the square into two right-angled triangles. If the side of the square is 's', 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:**
   - 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 $\frac{a^2}{2} \text{ cm}^2$.

3. **Justification for Option 3:**
   - Option 3 states that the area of the square is $\frac{a^2}{2} \text{ cm}^2$, which matches our calculated area. Thus, Option 3 is correct.

4. **Explanation of Incorrect Options:**
   - **Option 1: $2a \text{ cm}^2$**
     - This option suggests that the area is directly proportional to the diagonal, which is incorrect. The area should be proportional to the square of the side, not the diagonal.
   - **Option 2: $\frac{a}{\sqrt{2}} \text{ cm}^2$**
     - This option incorrectly suggests that the area is a linear function of the diagonal divided by $\sqrt{2}$. This is actually the expression for the side length, not the area.
   - **Option 4: $\frac{a^2}{4} \text{ cm}^2$**
     - This option suggests that the area is half of what it should be. The correct expression for the area is $\frac{a^2}{2}$, not $\frac{a^2}{4}$.

In conclusion, Option 3 is the correct answer because it accurately represents the area of the square when the diagonal is 'a' cm.