In `DeltaABC , angleB` is right angle and D is the mid-point of the side AC . If AB = 6 cm , BC = 8 cm , then the length of BD is

Options

A. 4 cm
B. 5 cm
C. 8 cm
D. 12 cm

Answer

To solve the problem, we need to find the length of segment BD in triangle ABC, where angle B is a right angle, D is the midpoint of side AC, AB = 6 cm, and BC = 8 cm.

### Step-by-Step Solution:

1. **Identify the Triangle and Given Values**:
- We have triangle ABC with angle B as a right angle.
- Given: AB = 6 cm, BC = 8 cm.

2. **Apply the Pythagorean Theorem**:
- In a right triangle, the Pythagorean theorem states that \( AC^2 = AB^2 + BC^2 \).
- Calculate \( AC^2 \):
\[ AC^2 = AB^2 + BC^2 = 6^2 + 8^2 = 36 + 64 = 100 \]
- Therefore, \( AC = \sqrt{100} = 10 \) cm.

3. **Determine the Length of AD and DC**:
- Since D is the midpoint of AC, we have:
\[ AD = DC = \frac{AC}{2} = \frac{10}{2} = 5 \text{ cm} \]

4. **Apply the Median Length Formula**:
- In a right triangle, the median to the hypotenuse is half the length of the hypotenuse. Here, BD is the median to hypotenuse AC.
- Therefore, \( BD = \frac{1}{2} AC = \frac{1}{2} \times 10 = 5 \text{ cm} \).

5. **Conclusion**:
- The length of BD is 5 cm.

### Final Answer:
The length of BD is **5 cm**.