In a triangle, with usual notations, the length of the bisector of angle A is

Options

A. `(2bc cosA/2)/(b+c)`
B. `(2bc sinA/2)/(b+c)`
C. `(abc " cosec " A/2)/(2R(b+c))`
D. `(2Delta)/(b+c) " cosec " A/2`

Answer

To find the length of the bisector of angle A in a triangle ABC, we can use the formula for the length of an angle bisector. The formula is given by:

\[ l_a = \frac{2bc}{b+c} \cdot \cos\left(\frac{A}{2}\right) \]

where:
- \( l_a \) is the length of the angle bisector from vertex A,
- \( b \) and \( c \) are the lengths of the sides opposite to vertices B and C respectively,
- \( A \) is the angle at vertex A.

### Step-by-Step Solution:

1. **Identify the sides and angle**:
In triangle ABC, identify the lengths of sides \( b \) and \( c \) opposite to angles B and C respectively, and the angle \( A \) at vertex A.

2. **Apply the angle bisector formula**:
Use the formula for the length of the angle bisector:
\[ l_a = \frac{2bc}{b+c} \cdot \cos\left(\frac{A}{2}\right) \]

3. **Calculate \( \cos\left(\frac{A}{2}\right) \)**:
If necessary, calculate \( \cos\left(\frac{A}{2}\right) \) using trigonometric identities or tables.

4. **Substitute the values**:
Substitute the values of \( b \), \( c \), and \( \cos\left(\frac{A}{2}\right) \) into the formula.

5. **Simplify the expression**:
Simplify the expression to find the length of the angle bisector \( l_a \).

### Example Calculation:
Assuming \( A = 60^\circ \), \( b = 5 \), and \( c = 7 \):
1. Calculate \( \cos\left(\frac{60^\circ}{2}\right) = \cos(30^\circ) = \frac{\sqrt{3}}{2} \).
2. Substitute into the formula:
\[ l_a = \frac{2 \cdot 5 \cdot 7}{5 + 7} \cdot \frac{\sqrt{3}}{2} \]
3. Simplify:
\[ l_a = \frac{70}{12} \cdot \frac{\sqrt{3}}{2} = \frac{35\sqrt{3}}{12} \]

### Final Result:
The length of the bisector of angle A is \( \frac{35\sqrt{3}}{12} \).