If the length of a diagonal of a square is 8 cm. Find the length of the side of the square.

Answer

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

### Step-by-Step Solution:

1. **Understand the Properties of a Square**:
- A square has four equal sides and its diagonals bisect each other at right angles (90 degrees).

2. **Identify the Given Information**:
- The length of the diagonal (AC) is given as 8 cm.

3. **Apply the Pythagorean Theorem**:
- In a square, if we denote the length of each side as 'A', then the diagonal can be calculated using the Pythagorean theorem:
\[ AC^2 = AB^2 + BC^2 \]
- Here, \( AC \) is the diagonal, \( AB \) and \( BC \) are the sides of the square. Since both sides are equal, we can write:
\[ AC^2 = A^2 + A^2 = 2A^2 \]

4. **Substitute the Value of the Diagonal**:
- We know that \( AC = 8 \) cm, so:
\[ 8^2 = 2A^2 \]
- This simplifies to:
\[ 64 = 2A^2 \]

5. **Solve for A²**:
- Divide both sides by 2:
\[ A^2 = \frac{64}{2} = 32 \]

6. **Find the Length of the Side (A)**:
- To find A, take the square root of both sides:
\[ A = \sqrt{32} \]
- We can simplify \( \sqrt{32} \):
\[ A = \sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2} = 4\sqrt{2} \]

7. **Final Answer**:
- The length of the side of the square is \( 4\sqrt{2} \) cm.