The length of the diagonal of square is a cm . Which of the following represents the area of the square (in sq cm ) ?

Options

A. `2a`
B. `1/sqrt(2)`
C. `a^2 // 2`
D. `a^2 // 4`

Answer

To find the area of a square when the length of its diagonal is given, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between the diagonal and the side of a square**:
The diagonal \(d\) of a square is related to its side length \(s\) by the formula:
\[ d = s \sqrt{2} \]
where \(d\) is the length of the diagonal and \(s\) is the length of the side.

2. **Substitute the given diagonal length**:
In this case, the diagonal is given as \(a\) cm. Therefore, we can write:
\[ a = s \sqrt{2} \]

3. **Solve for the side length \(s\)**:
To find the side length \(s\), we rearrange the equation:
\[ s = \frac{a}{\sqrt{2}} \]

4. **Find the area of the square**:
The area \(A\) of a square is given by the formula:
\[ A = s^2 \]
Substituting the expression for \(s\):
\[ A = \left(\frac{a}{\sqrt{2}}\right)^2 \]

5. **Simplify the area expression**:
Now simplify the expression:
\[ A = \frac{a^2}{2} \]

### Final Result:
The area of the square in square centimeters is:
\[ A = \frac{a^2}{2} \text{ sq cm} \]