If `tan A/2=5/6,Tan B/2=20/37.` Then sides are in

Options

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

Answer

To solve the problem, we need to find the relationship between the angles \( A \), \( B \), and \( C \) of a triangle based on the given values of \( \tan \frac{A}{2} \) and \( \tan \frac{B}{2} \).

### Step-by-Step Solution:

1. **Given Values**:
\[ \tan \frac{A}{2} = \frac{5}{6}, \quad \tan \frac{B}{2} = \frac{20}{37} \]

2. **Using the Triangle Angle Sum Property**:
Since \( A + B + C = 180^\circ \), we can express \( A + B \) as:
\[ A + B = 180^\circ - C \]
Dividing by 2 gives:
\[ \frac{A + B}{2} = 90^\circ - \frac{C}{2} \]

3. **Applying the Tangent Addition Formula**:
We can use the tangent addition formula:
\[ \tan\left(\frac{A + B}{2}\right) = \tan\left(90^\circ - \frac{C}{2}\right) = \cot\left(\frac{C}{2} \]
Thus, we have:
\[ \tan\left(\frac{A + B}{2}\right) = \frac{\tan \frac{A}{2} + \tan \frac{B}{2}}{1 - \tan \frac{A}{2} \tan \frac{B}{2}} \]

4. **Substituting the Values**:
Substitute the values of \( \tan \frac{A}{2} \) and \( \tan \frac{B}{2} \):
\[ \tan\left(\frac{A + B}{2}\right) = \frac{\frac{5}{6} + \frac{20}{37}}{1 - \left(\frac{5}{6} \cdot \frac{20}{37}\right)} \]

5. **Calculating the Numerator**:
To add \( \frac{5}{6} \) and \( \frac{20}{37} \), we find a common denominator:
\[ \text{Common denominator} = 6 \times 37 = 222 \]
Thus,
\[ \frac{5}{6} = \frac{5 \times 37}{222} = \frac{185}{222}, \quad \frac{20}{37} = \frac{20 \times 6}{222} = \frac{120}{222} \]
Therefore,
\[ \tan\left(\frac{A + B}{2}\right) = \frac{185 + 120}{222} = \frac{305}{222} \]

6. **Calculating the Denominator**:
Now calculate the denominator:
\[ 1 - \left(\frac{5}{6} \cdot \frac{20}{37}\right) = 1 - \frac{100}{222} = \frac{222 - 100}{222} = \frac{122}{222} \]

7. **Final Calculation**:
Now we can substitute back:
\[ \tan\left(\frac{A + B}{2}\right) = \frac{\frac{305}{222}}{\frac{122}{222}} = \frac{305}{122} \]

8. **Finding \( \tan \frac{C}{2} \)**:
Since \( \tan\left(\frac{A + B}{2}\right) = \cot\left(\frac{C}{2}\right) \), we have:
\[ \cot\left(\frac{C}{2}\right) = \frac{305}{122} \implies \tan\left(\frac{C}{2}\right) = \frac{122}{305} \]

9. **Comparing the Values**:
Now we compare:
- \( \tan \frac{A}{2} = \frac{5}{6} \approx 0.833 \)
- \( \tan \frac{B}{2} = \frac{20}{37} \approx 0.54 \)
- \( \tan \frac{C}{2} = \frac{122}{305} \approx 0.4 \)

From the comparisons:
\[ \tan \frac{A}{2} > \tan \frac{B}{2} > \tan \frac{C}{2} \]

10. **Conclusion**:
Since \( \tan \frac{A}{2} > \tan \frac{B}{2} > \tan \frac{C}{2} \), we conclude:
\[ A > B > C \]
Therefore, the sides opposite to these angles will also follow the same order:
\[ a > b > c \]

### Final Answer:
The sides are in the order \( a > b > c \).