In a triangle ABC if sides AB = c = 4 cm, side AC = b = 6 cm and BC = a = 7, then answer the following questions
If AD is bisector of angle A then length of BD is

Options

A. `16/5`
B. `21/5`
C. `12/5`
D. `14/5`

Answer

To find the length of segment BD in triangle ABC, where AD is the angle bisector of angle A, we can use the Angle Bisector Theorem. The theorem states that the ratio of the two segments created by the angle bisector on the opposite side is equal to the ratio of the other two sides of the triangle.

### Step-by-step Solution:

1. **Identify the sides of the triangle**:
- AB (c) = 4 cm
- AC (b) = 6 cm
- BC (a) = 7 cm

2. **Set up the Angle Bisector Theorem**:
According to the theorem:
\[ \frac{AB}{AC} = \frac{BD}{DC} \]
This can be rewritten as:
\[ \frac{c}{b} = \frac{BD}{DC} \]

3. **Substituting the known values**:
Substitute the values of AB and AC:
\[ \frac{4}{6} = \frac{BD}{DC} \]
Simplifying the left side gives:
\[ \frac{2}{3} = \frac{BD}{DC} \]

4. **Express DC in terms of BD**:
Let BD = x. Then, DC can be expressed as:
\[ DC = 7 - x \]
Now substitute this into the ratio:
\[ \frac{2}{3} = \frac{x}{7 - x} \]

5. **Cross-multiply to solve for x**:
Cross-multiplying gives:
\[ 2(7 - x) = 3x \]
Expanding this:
\[ 14 - 2x = 3x \]

6. **Combine like terms**:
Adding 2x to both sides:
\[ 14 = 5x \]

7. **Solve for x**:
Dividing both sides by 5:
\[ x = \frac{14}{5} \text{ cm} \]

8. **Conclusion**:
Therefore, the length of BD is:
\[ BD = \frac{14}{5} \text{ cm} \text{ or } 2.8 \text{ cm} \]