In a `Delta ABC, B = pi//8 and C = 5pi //8 and ` the atitude `AD=h.` Then h:a is equal to

Options

A. `2:1`
B. `3:1`
C. `1:3`
D. `1:2`

Answer

To solve the problem, we need to find the ratio of the altitude \( h \) from vertex \( A \) to side \( BC \) in triangle \( ABC \) to the length of side \( a \) (which is the length of \( BC \)). Given the angles \( B = \frac{\pi}{8} \) and \( C = \frac{5\pi}{8} \), we can find angle \( A \) and then apply the sine rule.

### Step-by-Step Solution:

1. **Find Angle A**:
\[ A = \pi - B - C = \pi - \frac{\pi}{8} - \frac{5\pi}{8} = \pi - \frac{6\pi}{8} = \pi - \frac{3\pi}{4} = \frac{\pi}{4} \]

**Hint**: Use the property that the sum of angles in a triangle is \( \pi \).

2. **Apply the Sine Rule**:
The sine rule states that:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]
Here, we need to express \( h \) in terms of \( a \). The altitude \( h \) can be expressed as:
\[ h = b \cdot \sin C = c \cdot \sin B \]

3. **Expressing \( h \) in terms of \( a \)**:
From the sine rule:
\[ b = \frac{a \cdot \sin B}{\sin A} \quad \text{and} \quad c = \frac{a \cdot \sin C}{\sin A} \]
Therefore:
\[ h = b \cdot \sin C = \left(\frac{a \cdot \sin B}{\sin A}\right) \cdot \sin C \]

4. **Calculate \( \sin A, \sin B, \sin C \)**:
- \( \sin A = \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2} \)
- \( \sin B = \sin \frac{\pi}{8} = \frac{\sqrt{2 - \sqrt{2}}}{2} \)
- \( \sin C = \sin \frac{5\pi}{8} = \sin\left(\frac{\pi}{2} - \frac{\pi}{8}\right) = \cos \frac{\pi}{8} = \frac{\sqrt{2 + \sqrt{2}}}{2} \)

5. **Substituting back into the equation for \( h \)**:
\[ h = \left(\frac{a \cdot \frac{\sqrt{2 - \sqrt{2}}}{2}}{\frac{\sqrt{2}}{2}}\right) \cdot \frac{\sqrt{2 + \sqrt{2}}}{2} \]
Simplifying this gives:
\[ h = a \cdot \frac{\sqrt{2 - \sqrt{2}} \cdot \sqrt{2 + \sqrt{2}}}{2} \]

6. **Simplifying the expression**:
Using the identity \( \sqrt{2 - \sqrt{2}} \cdot \sqrt{2 + \sqrt{2}} = \sqrt{(2 - \sqrt{2})(2 + \sqrt{2})} = \sqrt{4 - 2} = \sqrt{2} \):
\[ h = a \cdot \frac{\sqrt{2}}{2} \]

7. **Finding the ratio \( \frac{h}{a} \)**:
\[ \frac{h}{a} = \frac{\sqrt{2}}{2} \]
Therefore, the ratio \( h:a \) is:
\[ h:a = 1:\sqrt{2} \]

### Final Ratio:
To express it in a simpler form, we can multiply both sides by \( \sqrt{2} \):
\[ h:a = 1:2 \]

### Conclusion:
Thus, the ratio \( h:a \) is \( 1:2 \).