The angle of elevation of the top of a hill from each of the verticles A,B,C of a horizontal triangle is `alpha`. The height of the hill is

Options

A. b tan `alpha` cosec `beta`
B. `1/2` a tan `alpha` cosec `alpha`
C. `1/2c tan alpha cosec beta`
D. none of these

Explanation

To solve the problem, we need to find the height of the hill given the angle of elevation from the vertices of a triangle. Let's denote the vertices of the triangle as A, B, and C, and the height of the hill as H.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a horizontal triangle ABC.
- The angle of elevation to the top of the hill from each vertex A, B, and C is denoted as α.

**Hint**: Visualize the triangle and the hill. The hill's height will create a right triangle with each vertex of the triangle.

2. **Define the Height and Radius**:
- Let the height of the hill be H.
- The distance from the center of the triangle to each vertex (which we will denote as R) can be derived from the circumradius formula.

**Hint**: Remember that the circumradius R can be calculated using the sides of the triangle and the angles.

3. **Using Trigonometry**:
- From vertex A, we can write the tangent of the angle of elevation:
\[ \tan(\alpha) = \frac{H}{R} \]
- Rearranging gives:
\[ H = R \cdot \tan(\alpha) \]

**Hint**: Recall that the tangent function relates the opposite side (height of the hill) to the adjacent side (distance to the hill).

4. **Finding the Radius (R)**:
- The circumradius R can be expressed in terms of the sides of the triangle:
\[ R = \frac{a}{2 \sin(\alpha)} \]
- Here, a is the length of the side opposite to vertex A.

**Hint**: The circumradius formula is crucial for connecting the triangle's dimensions with the angles.

5. **Substituting R into the Height Formula**:
- Substitute the expression for R back into the height equation:
\[ H = \left(\frac{a}{2 \sin(\alpha)}\right) \cdot \tan(\alpha) \]
- This simplifies to:
\[ H = \frac{a \cdot \tan(\alpha)}{2 \sin(\alpha)} \]

**Hint**: Use the identity \( \tan(\alpha) = \frac{\sin(\alpha)}{\cos(\alpha)} \) to simplify further if needed.

6. **Final Expression for Height**:
- Thus, the height of the hill can be expressed as:
\[ H = \frac{a}{2 \cos(\alpha)} \]

**Hint**: This final formula gives you the height in terms of the side length and the angle of elevation.

### Conclusion:
The height of the hill (H) can be calculated using the formula:
\[ H = \frac{a \cdot \tan(\alpha)}{2 \sin(\alpha)} \]
or simplified to:
\[ H = \frac{a}{2 \cos(\alpha)} \]