The length of the diagonal of a square is 12 cm. Find its area (in `cm^(2)`).

Options

A. 36
B. 72
C. 144
D. 48

Answer

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

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

### Step 2: Substitute the given diagonal length into the formula.
We know the diagonal \( d \) is 12 cm. Therefore, we can set up the equation:
\[ 12 = s \sqrt{2} \]

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

### Step 4: Rationalize the denominator.
To rationalize \( \frac{12}{\sqrt{2}} \), we multiply the numerator and denominator by \( \sqrt{2} \):
\[ s = \frac{12 \sqrt{2}}{2} = 6 \sqrt{2} \]

### Step 5: Calculate the area of the square.
The area \( A \) of a square is given by the formula:
\[ A = s^2 \]
Substituting \( s = 6 \sqrt{2} \):
\[ A = (6 \sqrt{2})^2 \]

### Step 6: Simplify the area calculation.
Calculating the square:
\[ A = 6^2 \cdot (\sqrt{2})^2 = 36 \cdot 2 = 72 \text{ cm}^2 \]

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

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