The length of the diagonal of a square is 20 cm. its area is

Options

A. `400cm^(2)`
B. `200cm^(2)`
C. `300cm^(2)`
D. `100sqrt(2)cm^(2)`

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 the square:**
The diagonal (d) of a square can be related to the side length (s) using the formula:
\[ d = s \sqrt{2} \]
Here, \(d\) is the length of the diagonal, and \(s\) is the length of one side of the square.

2. **Substitute the given diagonal length into the formula:**
Given that the diagonal \(d = 20 \, \text{cm}\), we can substitute this value into the formula:
\[ 20 = s \sqrt{2} \]

3. **Solve for the side length (s):**
To find \(s\), we rearrange the equation:
\[ s = \frac{20}{\sqrt{2}} \]
To simplify this, we can multiply the numerator and the denominator by \(\sqrt{2}\):
\[ s = \frac{20 \sqrt{2}}{2} = 10 \sqrt{2} \, \text{cm} \]

4. **Calculate the area of the square:**
The area \(A\) of a square is given by the formula:
\[ A = s^2 \]
Substituting the value of \(s\):
\[ A = (10 \sqrt{2})^2 = 100 \cdot 2 = 200 \, \text{cm}^2 \]

5. **Final Answer:**
Therefore, the area of the square is:
\[ \text{Area} = 200 \, \text{cm}^2 \]