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 9 ft and 16 ft. respectively are complementary angles. The height of the tower is

Options

A. 9 ft
B. 12 ft
C. 16 ft
D. 144 ft

Answer

To solve the problem, we will use the concept of complementary angles and the properties of right-angled triangles.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have a tower and two points (A and B) from which the angles of elevation to the top of the tower are complementary. The distances from the foot of the tower to points A and B are 9 ft and 16 ft, respectively.

2. **Define Variables**:
Let \( h \) be the height of the tower (CD). Let \( \theta \) be the angle of elevation from point A (9 ft away), then the angle of elevation from point B (16 ft away) will be \( 90^\circ - \theta \).

3. **Set Up the Right Triangle Relationships**:
From point A (9 ft away):
\[ \tan(\theta) = \frac{h}{9} \quad \text{(1)} \]

From point B (16 ft away):
\[ \tan(90^\circ - \theta) = \cot(\theta) = \frac{h}{16} \quad \text{(2)} \]

4. **Relate the Two Equations**:
From equation (1):
\[ h = 9 \tan(\theta) \]
From equation (2):
\[ h = 16 \cot(\theta) \]
Since \( \cot(\theta) = \frac{1}{\tan(\theta)} \), we can substitute:
\[ h = 16 \cdot \frac{1}{\tan(\theta)} \]

5. **Equate the Two Expressions for h**:
Setting the two expressions for \( h \) equal gives:
\[ 9 \tan(\theta) = 16 \cdot \frac{1}{\tan(\theta)} \]

6. **Cross Multiply**:
\[ 9 \tan^2(\theta) = 16 \]

7. **Solve for \( \tan^2(\theta) \)**:
\[ \tan^2(\theta) = \frac{16}{9} \]

8. **Find \( h \)**:
Substitute \( \tan(\theta) \) back into either expression for \( h \):
\[ h = 9 \tan(\theta) = 9 \cdot \sqrt{\frac{16}{9}} = 9 \cdot \frac{4}{3} = 12 \text{ ft} \]

### Conclusion:
The height of the tower is \( h = 12 \) ft.

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