The diagonal of a square is 15 m, find the length of its one side and perimeter.

Answer

To solve the problem of finding the length of one side and the perimeter of a square given that its diagonal is 15 m, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Relationship Between Diagonal and Side of a Square**:
The diagonal \( d \) of a square is related to its side length \( a \) by the formula:
\[ d = a\sqrt{2} \]
Here, \( d \) is the length of the diagonal, and \( a \) is the length of one side of the square.

2. **Substituting the Given Diagonal**:
We know the diagonal \( d = 15 \) m. We can substitute this value into the formula:
\[ 15 = a\sqrt{2} \]

3. **Solving for the Side Length \( a \)**:
To find \( a \), we can rearrange the equation:
\[ a = \frac{15}{\sqrt{2}} \]

4. **Rationalizing the Denominator**:
To make the answer neater, we can rationalize the denominator:
\[ a = \frac{15}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{15\sqrt{2}}{2} \]
Thus, the length of one side of the square is \( \frac{15\sqrt{2}}{2} \) m.

5. **Calculating the Perimeter**:
The perimeter \( P \) of a square is given by the formula:
\[ P = 4a \]
Substituting the value of \( a \):
\[ P = 4 \cdot \frac{15\sqrt{2}}{2} = 30\sqrt{2} \text{ m} \]

### Final Answers:
- Length of one side of the square: \( \frac{15\sqrt{2}}{2} \) m
- Perimeter of the square: \( 30\sqrt{2} \) m