In a triangle ABC the length of the bisector of angle A is :
(i) `2(bc)/(b+c) sin.(A)/(2)`
(ii) `2(bc)/(b+c) cos.(A)/(2)`
(iii) `(abc)/(2R(b+c)) cosec.(A)/(2)`
(iv) `(4A)/(b+c) cosec.(A)/(2)`

Options

A. (i), (ii)
B. (ii), (iii)
C. (ii), (iv)
D. (i), (iv)

Answer

To find the length of the angle bisector \( AD \) in triangle \( ABC \), we can use the angle bisector theorem and the sine rule. Here’s a step-by-step solution:

### Step 1: Apply the Angle Bisector Theorem
According to the angle bisector theorem, the angle bisector divides the opposite side in the ratio of the adjacent sides. Therefore, we have:

\[ \frac{BD}{DC} = \frac{c}{b} \]

Let \( BD = x \) and \( DC = y \). Then, we can express \( x \) and \( y \) in terms of \( A \):

\[ x + y = a \quad \text{(where \( a = BC \))} \]

From the angle bisector theorem, we can write:

\[ \frac{x}{y} = \frac{c}{b} \implies x = \frac{c}{b+c}a \quad \text{and} \quad y = \frac{b}{b+c}a \]

### Step 2: Use the Sine Rule
Now, we apply the sine rule in triangle \( ABD \):

\[ \frac{AD}{\sin B} = \frac{BD}{\sin A/2} \]

Substituting \( BD \) from the previous step:

\[ \frac{AD}{\sin B} = \frac{\frac{c}{b+c}a}{\sin A/2} \]

### Step 3: Solve for \( AD \)
Rearranging the equation gives:

\[ AD = \frac{c \cdot a \cdot \sin B}{(b+c) \cdot \sin A/2} \]

### Step 4: Substitute \( a \) using the Sine Rule
Using the sine rule in triangle \( ABC \):

\[ \frac{a}{\sin A} = 2R \implies a = 2R \sin A \]

Substituting \( a \) back into the equation for \( AD \):

\[ AD = \frac{c \cdot 2R \sin A \cdot \sin B}{(b+c) \cdot \sin A/2} \]

### Step 5: Simplify the Expression
Now, we can simplify the expression further. Using the double angle identity:

\[ \sin A = 2 \sin \frac{A}{2} \cos \frac{A}{2} \]

Thus, we have:

\[ AD = \frac{2cR \cdot 2 \sin \frac{A}{2} \cos \frac{A}{2} \cdot \sin B}{(b+c) \cdot \sin \frac{A}{2}} \]

Cancelling \( \sin \frac{A}{2} \):

\[ AD = \frac{4cR \cos \frac{A}{2} \sin B}{b+c} \]

### Step 6: Final Form
Now we can express \( AD \) in terms of \( a, b, c, R \) and \( A \):

\[ AD = \frac{abc}{2R(b+c)} \cdot \csc \frac{A}{2} \]

### Conclusion
Thus, the correct option for the length of the bisector of angle \( A \) is:

**Option (iii)**: \( \frac{abc}{2R(b+c)} \csc \frac{A}{2} \)

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