If AD is the bisector of `angleBAC` and AB = 10 cm, AC = 6 cm, BC = 12 cm, then find BD?
यदि AD, `angleBAC` का समद्विभाजक है तथा AB = 10cm, AC = 6cm, BC = 12cm, तो BD ज्ञात करें।

Options

A. 6 cm
B. 7.5 cm
C. 8 cm
D. 8.5 cm

Explanation

To find the length of BD when AD is the bisector of angle BAC in triangle ABC, we can use the Angle Bisector Theorem. According to this theorem, the angle bisector divides the opposite side into segments that are proportional to the lengths of the other two sides.

### Step-by-Step Solution:

1. **Identify the given lengths:**
- AB = 10 cm
- AC = 6 cm
- BC = 12 cm

2. **Apply the Angle Bisector Theorem:**
The Angle Bisector Theorem states that:
\[ \frac{BD}{DC} = \frac{AB}{AC} \]
Let BD = x and DC = y. Since BC = BD + DC, we have:
\[ x + y = 12 \quad (1) \]

3. **Set up the ratio using the theorem:**
From the theorem:
\[ \frac{x}{y} = \frac{10}{6} = \frac{5}{3} \]
This implies:
\[ 3x = 5y \quad (2) \]

4. **Substitute equation (1) into equation (2):**
From equation (1), we can express y in terms of x:
\[ y = 12 - x \]
Substitute this into equation (2):
\[ 3x = 5(12 - x) \]
Simplifying this gives:
\[ 3x = 60 - 5x \]
\[ 3x + 5x = 60 \]
\[ 8x = 60 \]
\[ x = \frac{60}{8} = 7.5 \]

5. **Find y using equation (1):**
Substitute x back into equation (1):
\[ y = 12 - 7.5 = 4.5 \]

6. **Conclusion:**
Therefore, BD = x = 7.5 cm.

### Final Answer:
BD = 7.5 cm