In a triangle ABC if `tan.(A)/(2)tan.(B)/(2)=(1)/(3)` and ab = 4, then the value of c can be

Options

A. 1
B. `1.5`
C. `2.5`
D. none of these

Explanation

To solve the problem, we need to find the value of side \( c \) in triangle \( ABC \) given that:

\[ \frac{\tan \left( \frac{A}{2} \right) \tan \left( \frac{B}{2} \right)}{1} = \frac{1}{3} \]

and \( ab = 4 \).

### Step-by-Step Solution:

1. **Use the formula for \( \tan \left( \frac{A}{2} \right) \) and \( \tan \left( \frac{B}{2} \right) \)**:
\[ \tan \left( \frac{A}{2} \right) = \frac{s - b}{s} \quad \text{and} \quad \tan \left( \frac{B}{2} \right) = \frac{s - a}{s} \]
where \( s \) is the semi-perimeter of the triangle given by \( s = \frac{a + b + c}{2} \).

2. **Substituting the formulas into the equation**:
\[ \tan \left( \frac{A}{2} \right) \tan \left( \frac{B}{2} \right) = \frac{(s - b)(s - a)}{s^2} \]
Therefore, we have:
\[ \frac{(s - b)(s - a)}{s^2} = \frac{1}{3} \]

3. **Cross-multiply to eliminate the fraction**:
\[ 3(s - b)(s - a) = s^2 \]

4. **Expand the left-hand side**:
\[ 3(s^2 - (a + b)s + ab) = s^2 \]
This simplifies to:
\[ 3s^2 - 3(a + b)s + 3ab = s^2 \]

5. **Rearranging the equation**:
\[ 2s^2 - 3(a + b)s + 3ab = 0 \]

6. **Using the quadratic formula**:
The quadratic formula is given by:
\[ s = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \]
Here, \( A = 2 \), \( B = -3(a + b) \), and \( C = 3ab \).

7. **Substituting values**:
\[ s = \frac{3(a + b) \pm \sqrt{(-3(a + b))^2 - 4 \cdot 2 \cdot 3ab}}{2 \cdot 2} \]
Simplifying further leads to:
\[ s = \frac{3(a + b) \pm \sqrt{9(a + b)^2 - 24ab}}{4} \]

8. **Using the condition \( ab = 4 \)**:
Substitute \( ab = 4 \) into the equation:
\[ s = \frac{3(a + b) \pm \sqrt{9(a + b)^2 - 96}}{4} \]

9. **Finding the relationship between \( a + b \) and \( c \)**:
From the semi-perimeter, we know:
\[ a + b + c = 2s \]
Therefore:
\[ c = 2s - (a + b) \]

10. **Using the AM-GM inequality**:
The arithmetic mean-geometric mean inequality states:
\[ \frac{a + b}{2} \geq \sqrt{ab} \]
Thus:
\[ a + b \geq 2\sqrt{ab} = 2\sqrt{4} = 4 \]

11. **Substituting back to find \( c \)**:
From the earlier relationship:
\[ c \geq 2 \]

### Conclusion:
Thus, the value of \( c \) can be \( \geq 2 \).