In a `Delta ABC`, if `tan.(A)/(2)=(5)/(6), tan.(B)/(2)=(20)/(37)`, then which of the following is/are correct ?

Options

A. `angle B gt angle C`
B. `angle B lt angle C`
C. `a gt b gt c`
D. `a lt b lt c`

Explanation

To solve the problem, we will use the properties of tangent and the relationships between the angles and sides of triangle ABC.

### Step-by-Step Solution:

1. **Given Information**:
We have:
\[ \tan\left(\frac{A}{2}\right) = \frac{5}{6} \]
\[ \tan\left(\frac{B}{2}\right) = \frac{20}{37} \]

2. **Using the Tangent Addition Formula**:
We know that:
\[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]
We can express \( \tan A \) and \( \tan B \) in terms of \( \tan\left(\frac{A}{2}\right) \) and \( \tan\left(\frac{B}{2}\right) \):
\[ \tan A = \frac{2 \tan\left(\frac{A}{2}\right)}{1 - \tan^2\left(\frac{A}{2}\right)} \]
\[ \tan B = \frac{2 \tan\left(\frac{B}{2}\right)}{1 - \tan^2\left(\frac{B}{2}\right)} \]

3. **Calculating \( \tan A \) and \( \tan B \)**:
First, calculate \( \tan A \):
\[ \tan A = \frac{2 \cdot \frac{5}{6}}{1 - \left(\frac{5}{6}\right)^2} = \frac{\frac{10}{6}}{1 - \frac{25}{36}} = \frac{\frac{10}{6}}{\frac{11}{36}} = \frac{10 \cdot 36}{6 \cdot 11} = \frac{60}{11} \]

Now, calculate \( \tan B \):
\[ \tan B = \frac{2 \cdot \frac{20}{37}}{1 - \left(\frac{20}{37}\right)^2} = \frac{\frac{40}{37}}{1 - \frac{400}{1369}} = \frac{\frac{40}{37}}{\frac{969}{1369}} = \frac{40 \cdot 1369}{37 \cdot 969} = \frac{54760}{35853} \]

4. **Finding \( \tan(A + B) \)**:
Now we can find \( \tan(A + B) \):
\[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]
Substitute the values of \( \tan A \) and \( \tan B \) into this formula.

5. **Using the Angle Sum Property**:
Since \( A + B + C = 180^\circ \), we have:
\[ A + B = 180^\circ - C \]
Therefore:
\[ \tan(A + B) = \cot C \]

6. **Finding \( \tan C \)**:
From the relationship \( \tan C = \frac{1}{\tan(A + B)} \), we can find \( \tan C \).

7. **Comparing Angles**:
We can compare \( \tan\left(\frac{A}{2}\right) \), \( \tan\left(\frac{B}{2}\right) \), and \( \tan\left(\frac{C}{2}\right) \) to determine the order of angles:
- If \( \tan\left(\frac{A}{2}\right) > \tan\left(\frac{B}{2}\right) \), then \( A > B \).
- If \( \tan\left(\frac{B}{2}\right) > \tan\left(\frac{C}{2}\right) \), then \( B > C \).

8. **Conclusion**:
From the calculations, we can conclude the order of angles and subsequently the order of sides based on the properties of triangles.