The lengths of the diagonals of a rhombus are 16 cm and 12 cm respectively . Find the length of each of its sides.

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 lengths of the diagonals:**
- Let the lengths of the diagonals be \( AC = 16 \, \text{cm} \) and \( BD = 12 \, \text{cm} \).

2. **Find the lengths of half the diagonals:**
- Since the diagonals of a rhombus bisect each other at right angles, we can find \( AO \) and \( BO \):
- \( AO = \frac{AC}{2} = \frac{16}{2} = 8 \, \text{cm} \)
- \( BO = \frac{BD}{2} = \frac{12}{2} = 6 \, \text{cm} \)

3. **Use the Pythagorean theorem:**
- In triangle \( AOB \), we can apply the Pythagorean theorem:
\[ AB^2 = AO^2 + BO^2 \]
- Substitute the values:
\[ AB^2 = 8^2 + 6^2 \]
- Calculate \( 8^2 \) and \( 6^2 \):
\[ AB^2 = 64 + 36 \]
- Add the squares:
\[ AB^2 = 100 \]

4. **Find the length of side \( AB \):**
- Take the square root to find \( AB \):
\[ AB = \sqrt{100} = 10 \, \text{cm} \]

5. **Conclusion:**
- Since all sides of a rhombus are equal, the length of each side of the rhombus is \( 10 \, \text{cm} \).

### Final Answer:
The length of each side of the rhombus is \( 10 \, \text{cm} \).