The lengths of two diagonals of a rhombus are 6 cm and 8 cm. What is the side (in cm) of the rhombus?

Options

A. 5
B. 10
C. `5 sqrt2`
D. `10sqrt2`

Answer

To find the 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 \( D_1 = 6 \) cm and \( D_2 = 8 \) cm.

2. **Use the relationship between the diagonals and the side of the rhombus:**
- The relationship is given by the formula:
\[ D_1^2 + D_2^2 = 4A^2 \]
where \( A \) is the length of a side of the rhombus.

3. **Substitute the values of the diagonals into the formula:**
- Substitute \( D_1 \) and \( D_2 \):
\[ 6^2 + 8^2 = 4A^2 \]

4. **Calculate the squares of the diagonals:**
- Calculate \( 6^2 \) and \( 8^2 \):
\[ 6^2 = 36 \quad \text{and} \quad 8^2 = 64 \]
- Add these values:
\[ 36 + 64 = 100 \]

5. **Set up the equation:**
- Now we have:
\[ 100 = 4A^2 \]

6. **Solve for \( A^2 \):**
- Divide both sides by 4:
\[ A^2 = \frac{100}{4} = 25 \]

7. **Find \( A \):**
- Take the square root of both sides:
\[ A = \sqrt{25} = 5 \]

8. **Conclusion:**
- The length of each side of the rhombus is \( 5 \) cm.

### Final Answer:
The side of the rhombus is \( 5 \) cm.