The diagonals of a rhombus are 8 cm and 6 cm, then the length of each side of the rhombus is

Answer

To find the length of each side of the rhombus given the lengths of its diagonals, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the diagonals**:
Let the diagonals of the rhombus be \( AC \) and \( BD \). Given that \( AC = 8 \, \text{cm} \) and \( BD = 6 \, \text{cm} \).

2. **Find the half-lengths of the diagonals**:
Since the diagonals of a rhombus bisect each other at right angles, we can find the lengths of the segments formed by the intersection point \( O \):
- \( AO = \frac{AC}{2} = \frac{8}{2} = 4 \, \text{cm} \)
- \( OB = \frac{BD}{2} = \frac{6}{2} = 3 \, \text{cm} \)

3. **Use the Pythagorean theorem**:
In triangle \( AOB \), we can apply the Pythagorean theorem since \( \angle AOB = 90^\circ \):
\[ AB^2 = AO^2 + OB^2 \]
Substituting the values we found:
\[ AB^2 = 4^2 + 3^2 = 16 + 9 = 25 \]

4. **Calculate the length of side \( AB \)**:
Taking the square root of both sides:
\[ AB = \sqrt{25} = 5 \, \text{cm} \]

5. **Conclusion**:
Since all sides of a rhombus are equal, the length of each side of the rhombus is \( 5 \, \text{cm} \).

### Final Answer:
The length of each side of the rhombus is \( 5 \, \text{cm} \).

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