In a `Delta ABC , angle A=120^(@), AB=20 cm, AC =30 cm` find the angle bisector AD .

Answer

To solve the problem of finding the length of the angle bisector \( AD \) in triangle \( ABC \) where \( \angle A = 120^\circ \), \( AB = 20 \, \text{cm} \), and \( AC = 30 \, \text{cm} \), we can use the angle bisector theorem and the formula for the length of an angle bisector.

### Step-by-Step Solution:

1. **Identify the given values:**
- \( AB = c = 20 \, \text{cm} \)
- \( AC = b = 30 \, \text{cm} \)
- \( \angle A = 120^\circ \)

2. **Use the angle bisector theorem:**
The angle bisector theorem states that the ratio of the lengths of the two segments created by the angle bisector on the opposite side is equal to the ratio of the other two sides.
\[ \frac{BD}{DC} = \frac{AB}{AC} = \frac{c}{b} = \frac{20}{30} = \frac{2}{3} \]

3. **Let \( BD = 2x \) and \( DC = 3x \):**
Since \( BD + DC = BC \), we can express \( BC \) as:
\[ BC = BD + DC = 2x + 3x = 5x \]

4. **Use the formula for the length of the angle bisector:**
The length of the angle bisector \( AD \) can be calculated using the formula:
\[ AD = \frac{2bc}{b+c} \cdot \cos\left(\frac{A}{2}\right) \]
where \( A \) is the angle at vertex \( A \), \( b \) and \( c \) are the lengths of the sides opposite to angles \( B \) and \( C \) respectively.

5. **Calculate \( \cos\left(\frac{A}{2}\right) \):**
Since \( A = 120^\circ \):
\[ \frac{A}{2} = 60^\circ \quad \text{and} \quad \cos(60^\circ) = \frac{1}{2} \]

6. **Substitute the values into the formula:**
\[ AD = \frac{2 \cdot 30 \cdot 20}{30 + 20} \cdot \cos(60^\circ) \]
\[ AD = \frac{1200}{50} \cdot \frac{1}{2} \]
\[ AD = 24 \cdot \frac{1}{2} = 12 \, \text{cm} \]

### Final Answer:
The length of the angle bisector \( AD \) is \( 12 \, \text{cm} \).