The lengths of the diagonals of a rhombus are 16 cm and 12 cm . The length of each side of the rhombus is

Options

A. 8cm
B. 9 cm
C. 10 cm
D. 12 cm

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 \( d_1 = 16 \) cm and \( d_2 = 12 \) 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 half-lengths:
- Half of \( d_1 \): \( \frac{d_1}{2} = \frac{16}{2} = 8 \) cm
- Half of \( d_2 \): \( \frac{d_2}{2} = \frac{12}{2} = 6 \) cm

3. **Visualize the right triangle**:
- The diagonals intersect at point O, forming four right triangles within the rhombus. We can focus on one of these triangles, say triangle OBC, where:
- OB = 8 cm (half of diagonal BD)
- OC = 6 cm (half of diagonal AC)

4. **Apply the Pythagorean theorem**:
- In triangle OBC, we can use the Pythagorean theorem to find the length of side BC (which is also the length of each side of the rhombus):
\[ BC^2 = OB^2 + OC^2 \]
\[ BC^2 = 8^2 + 6^2 \]

5. **Calculate the squares**:
- Calculate \( 8^2 \) and \( 6^2 \):
\[ 8^2 = 64 \]
\[ 6^2 = 36 \]

6. **Sum the squares**:
- Now, add the two results:
\[ BC^2 = 64 + 36 = 100 \]

7. **Find the length of BC**:
- To find BC, take the square root of both sides:
\[ BC = \sqrt{100} = 10 \text{ cm} \]

8. **Conclusion**:
- Therefore, the length of each side of the rhombus is \( 10 \) cm.

### Final Answer:
The length of each side of the rhombus is **10 cm**.