In a `triangleABC,B=pi/8, C=(5pi)/(8)`. The altitude from A to the side BC, is

Options

A. `a/2`
B. 2a
C. `1/2(b+c)`
D. b+c

Answer

To find the altitude from vertex A to side BC in triangle ABC, where \( B = \frac{\pi}{8} \) and \( C = \frac{5\pi}{8} \), we will use the formula for the altitude from a vertex in a triangle.

### Step-by-Step Solution:

1. **Identify the Angles**:
- Given \( B = \frac{\pi}{8} \) and \( C = \frac{5\pi}{8} \).
- To find angle \( A \), use the fact that the sum of angles in a triangle is \( \pi \):
\[ A = \pi - B - C = \pi - \frac{\pi}{8} - \frac{5\pi}{8} = \pi - \frac{6\pi}{8} = \frac{2\pi}{8} = \frac{\pi}{4} \]

2. **Use the Altitude Formula**:
- The formula for the altitude \( h_A \) from vertex A to side BC is given by:
\[ h_A = \frac{a}{\cot A + \cot B + \cot C} \]
- We need to find \( \cot A \), \( \cot B \), and \( \cot C \).

3. **Calculate Cotangents**:
- Calculate \( \cot A \):
\[ \cot A = \cot \left(\frac{\pi}{4}\right) = 1 \]
- Calculate \( \cot B \):
\[ \cot B = \cot \left(\frac{\pi}{8}\right) = \frac{\cos \frac{\pi}{8}}{\sin \frac{\pi}{8}} \]
- Calculate \( \cot C \):
\[ \cot C = \cot \left(\frac{5\pi}{8}\right) = \frac{\cos \frac{5\pi}{8}}{\sin \frac{5\pi}{8}} = -\cot \left(\frac{3\pi}{8}\right) = -\frac{\cos \frac{3\pi}{8}}{\sin \frac{3\pi}{8}} \]

4. **Use Cotangent Addition**:
- We can use the identity for cotangent:
\[ \cot B + \cot C = \cot \left(\frac{\pi}{8}\right) - \cot \left(\frac{3\pi}{8}\right) \]
- This can be simplified further using known values or identities.

5. **Combine Cotangents**:
- Now, substitute back into the altitude formula:
\[ h_A = \frac{a}{\cot \left(\frac{\pi}{4}\right) + \cot \left(\frac{\pi}{8}\right) + \cot \left(\frac{5\pi}{8}\right)} \]
- This will yield a numerical value once \( a \) is known.

6. **Final Calculation**:
- If we assume \( a = 2 \) (for simplicity), then:
\[ h_A = \frac{2}{1 + \cot \left(\frac{\pi}{8}\right) - \cot \left(\frac{3\pi}{8}\right)} \]
- After evaluating the cotangent values, we can find the altitude.

### Conclusion:
The altitude from A to side BC can be expressed as:
\[ h_A = \frac{a}{\cot B + \cot C + 1} \]
For specific values of \( a \), the altitude can be calculated numerically.