A vertical pole stands at a point O on a horizontal ground. A and B are points on the ground d metres apart. The pole subtends angles `alpha and beta` at A and B respectively. AB subtends an angle at O. the height of the pole is ______.

Answer

To find the height of the vertical pole at point O, we can use the concept of trigonometry. Let's denote the height of the pole as \( h \).

### Step-by-Step Solution:

1. **Understanding the Geometry**:
- Let the height of the pole be \( h \).
- The distance from point A to the base of the pole (point O) is \( OA \).
- The distance from point B to the base of the pole (point O) is \( OB \).
- The distance between points A and B is given as \( d \).

2. **Using Trigonometric Ratios**:
- From point A, the angle subtended by the pole is \( \alpha \). According to the tangent function:
\[ \tan(\alpha) = \frac{h}{OA} \]
Rearranging gives:
\[ h = OA \cdot \tan(\alpha) \tag{1} \]

- From point B, the angle subtended by the pole is \( \beta \). Similarly, we have:
\[ \tan(\beta) = \frac{h}{OB} \]
Rearranging gives:
\[ h = OB \cdot \tan(\beta) \tag{2} \]

3. **Relating OA and OB**:
- Since A and B are \( d \) meters apart, we can express \( OA \) and \( OB \) in terms of \( d \):
\[ OA + OB = d \]
Let \( OA = x \) and \( OB = d - x \).

4. **Substituting in Equations**:
- Substitute \( OA \) and \( OB \) into equations (1) and (2):
\[ h = x \cdot \tan(\alpha) \tag{3} \]
\[ h = (d - x) \cdot \tan(\beta) \tag{4} \]

5. **Setting Equations Equal**:
- Since both expressions equal \( h \), we can set them equal to each other:
\[ x \cdot \tan(\alpha) = (d - x) \cdot \tan(\beta) \]

6. **Solving for x**:
- Rearranging gives:
\[ x \cdot \tan(\alpha) + x \cdot \tan(\beta) = d \cdot \tan(\beta) \]
\[ x(\tan(\alpha) + \tan(\beta)) = d \cdot \tan(\beta) \]
\[ x = \frac{d \cdot \tan(\beta)}{\tan(\alpha) + \tan(\beta)} \tag{5} \]

7. **Finding Height h**:
- Substitute \( x \) back into equation (3) to find \( h \):
\[ h = \left(\frac{d \cdot \tan(\beta)}{\tan(\alpha) + \tan(\beta)}\right) \cdot \tan(\alpha) \]
\[ h = \frac{d \cdot \tan(\alpha) \cdot \tan(\beta)}{\tan(\alpha) + \tan(\beta)} \]

### Final Answer:
The height of the pole \( h \) is given by:
\[ h = \frac{d \cdot \tan(\alpha) \cdot \tan(\beta)}{\tan(\alpha) + \tan(\beta)} \]