When a plane mirror is placed horizontally on level ground at a distance of 60 m from the foot of a tower, the top of the tower and its image in the mirror subtend and angle of `90^@` at the eye. The height of the tower is

Options

A. 30 m
B. 60 m
C. 90 m
D. 120 m

Explanation

To solve the problem, we need to find the height of the tower based on the information provided. Let's break it down step by step.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a tower (let's denote its height as \( h \)) and a plane mirror placed horizontally on the ground at a distance of 60 meters from the foot of the tower.
- The top of the tower and its image in the mirror subtend an angle of \( 90^\circ \) at the observer's eye.

2. **Identifying Key Points**:
- Let point A be the foot of the tower, point B be the top of the tower, and point M be the position of the mirror.
- The image of the top of the tower in the mirror will be directly below the top of the tower at a distance equal to the height of the tower.

3. **Visualizing the Angles**:
- Since the angle subtended at the eye by the top of the tower and its image is \( 90^\circ \), we can conclude that the angles formed at the eye (let's denote it as point E) are \( 45^\circ \) each (since \( 90^\circ \) is divided equally between the two angles).

4. **Using Right Triangle Properties**:
- In the right triangle \( ABE \) (where A is the foot of the tower, B is the top of the tower, and E is the observer's eye), we can use the tangent function:
\[ \tan(45^\circ) = \frac{h}{60} \]

5. **Calculating Height**:
- We know that \( \tan(45^\circ) = 1 \). Therefore, we can write:
\[ 1 = \frac{h}{60} \]
- Solving for \( h \):
\[ h = 60 \text{ meters} \]

6. **Conclusion**:
- The height of the tower is \( 60 \) meters.

### Final Answer:
The height of the tower is \( 60 \) meters.

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