The angle of elevation of the top of a tower standing on a horizontal plane from two points on a line passing through the foot of the tower at a distance x and y respectively are complementary angles. Find the height of the tower.

Answer

To solve the problem, we need to find the height of the tower given that the angles of elevation from two points on the ground are complementary. Let's denote the height of the tower as \( h \).

### Step-by-Step Solution:

1. **Understanding the Problem**:
- Let the foot of the tower be point \( A \).
- Let the top of the tower be point \( B \).
- Let the two points from which the angles of elevation are measured be points \( C \) and \( D \).
- The distances from point \( A \) to points \( C \) and \( D \) are \( x \) and \( y \) respectively.

2. **Setting Up the Angles**:
- Let the angle of elevation from point \( C \) be \( \theta \).
- Since the angles are complementary, the angle of elevation from point \( D \) will be \( 90^\circ - \theta \).

3. **Using Trigonometric Ratios**:
- From point \( C \):
\[ \tan(\theta) = \frac{h}{x} \quad \text{(1)} \]
- From point \( D \):
\[ \tan(90^\circ - \theta) = \cot(\theta) = \frac{h}{y} \quad \text{(2)} \]

4. **Relating the Two Equations**:
- From equation (1), we can express \( h \):
\[ h = x \tan(\theta) \quad \text{(3)} \]
- From equation (2), we can express \( h \) in terms of \( y \):
\[ h = y \cot(\theta) \quad \text{(4)} \]

5. **Setting Equations (3) and (4) Equal**:
- Since both expressions equal \( h \), we can set them equal to each other:
\[ x \tan(\theta) = y \cot(\theta) \]

6. **Using the Identity \( \cot(\theta) = \frac{1}{\tan(\theta)} \)**:
- Substitute \( \cot(\theta) \) in the equation:
\[ x \tan(\theta) = \frac{y}{\tan(\theta)} \]
- Multiplying both sides by \( \tan(\theta) \):
\[ x \tan^2(\theta) = y \]

7. **Solving for \( \tan^2(\theta) \)**:
- Rearranging gives:
\[ \tan^2(\theta) = \frac{y}{x} \]

8. **Finding the Height \( h \)**:
- Substitute \( \tan^2(\theta) \) back into equation (3):
\[ h = x \tan(\theta) = x \sqrt{\frac{y}{x}} = \sqrt{xy} \]

### Final Result:
The height of the tower \( h \) is given by:
\[ h = \sqrt{xy} \]