If the diagonal of a square is 12 cm, then what is the area (in `cm^(2)`) of the square?

Options

A. 72
B. `72 sqrt""2`
C. 36
D. `36sqrt""2`

Answer

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

### Step-by-Step Solution:

1. **Understand the relationship between the diagonal and the side of the square**:
The diagonal \( d \) of a square relates to its side length \( A \) through the formula:
\[ d = A \sqrt{2} \]

2. **Substitute the given diagonal value**:
We know the diagonal \( d \) is 12 cm. Therefore, we can set up the equation:
\[ 12 = A \sqrt{2} \]

3. **Solve for the side length \( A \)**:
To find \( A \), we can rearrange the equation:
\[ A = \frac{12}{\sqrt{2}} \]

4. **Rationalize the denominator**:
To simplify \( A \), we multiply the numerator and denominator by \( \sqrt{2} \):
\[ A = \frac{12 \sqrt{2}}{2} = 6 \sqrt{2} \text{ cm} \]

5. **Calculate the area of the square**:
The area \( \text{Area} \) of the square is given by the formula:
\[ \text{Area} = A^2 \]
Substituting \( A \):
\[ \text{Area} = (6 \sqrt{2})^2 \]

6. **Perform the squaring**:
\[ \text{Area} = 6^2 \cdot (\sqrt{2})^2 = 36 \cdot 2 = 72 \text{ cm}^2 \]

### Final Answer:
The area of the square is \( 72 \text{ cm}^2 \).

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