The angle of elevation of the top of a tower standing on a horizontal plane from two on a line passing through the foot of the tower at a distance 9 ft and 16 ft respectively are complementary angles. Then the height of the tower is:

Options

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

Answer

To solve the problem, we need to find the height of the tower given the distances from two points and the fact that the angles of elevation from these points are complementary.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- Let the height of the tower be \( h \).
- From point A (9 ft away from the tower), let the angle of elevation be \( \theta \).
- From point B (16 ft away from the tower), the angle of elevation will be \( 90^\circ - \theta \) since the angles are complementary.

2. **Setting Up the Right Triangles**:
- For point A:
\[ \tan(\theta) = \frac{h}{9} \]
- For point B:
\[ \tan(90^\circ - \theta) = \cot(\theta) = \frac{h}{16} \]

3. **Using the Relationship Between Tangent and Cotangent**:
- We know that:
\[ \cot(\theta) = \frac{1}{\tan(\theta)} \]
- Therefore, we can express \( \cot(\theta) \) in terms of \( h \):
\[ \cot(\theta) = \frac{9}{h} \]

4. **Setting the Equations Equal**:
- From the second equation:
\[ \frac{h}{16} = \frac{9}{h} \]
- Cross-multiplying gives:
\[ h^2 = 9 \times 16 \]

5. **Calculating \( h \)**:
- Simplifying the right side:
\[ h^2 = 144 \]
- Taking the square root:
\[ h = \sqrt{144} = 12 \]

6. **Conclusion**:
- The height of the tower is \( 12 \) feet.

### Final Answer:
The height of the tower is **12 feet**.

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