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 angle bisector theorem and some properties of triangles. The length of the angle bisector can be derived using the following formula:

### Step-by-Step Solution:

1. **Understand the Triangle and Notations**:
- Let triangle ABC have sides opposite to vertices A, B, and C denoted as a, b, and c respectively.
- The angle bisector of angle A divides the opposite side BC into two segments, which we can denote as D, where BD = m and DC = n.

2. **Apply the Angle Bisector Theorem**:
- According to the angle bisector theorem, the ratio of the lengths of the segments created by the angle bisector is equal to the ratio of the other two sides:
\[ \frac{m}{n} = \frac{b}{c} \]

3. **Express m and n in terms of a, b, and c**:
- Let \( m = \frac{b}{b+c} \cdot a \) and \( n = \frac{c}{b+c} \cdot a \). Thus, we can express the lengths of the segments created by the angle bisector in terms of the sides of the triangle.

4. **Use the Formula for the Length of the Angle Bisector**:
- The length of the angle bisector AD can be calculated using the formula:
\[ AD = \frac{2bc}{b+c} \cdot \cos\left(\frac{A}{2}\right) \]
- Here, \( A \) is the angle at vertex A.

5. **Substitute Values**:
- Substitute the values of b and c into the formula to find the length of the angle bisector AD.

6. **Final Result**:
- The length of the angle bisector AD is given by:
\[ AD = \frac{2bc \cdot \cos\left(\frac{A}{2}\right)}{b+c} \]