The figure shows an overhead view of a corridor with a plane mirror MN mounted at one end. A burglar B sneaks along the corridor directly towards the centre of the mirror. If d = 3m, how far from the mirror will the burglar be when the security guard S can first see him in the mirror ? ---

Options

A. 3.0m
B. 4.5m
C. 6.0m
D. 1.5m

Answer

To solve the problem, we need to determine how far the burglar B will be from the mirror when the security guard S can first see him in the mirror.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a corridor with a mirror at one end.
- The distance from the burglar to the mirror is denoted as \( x \).
- The total distance from the burglar to the point directly opposite the mirror (where the guard can see) is \( d = 3 \, m \).

2. **Identifying the Geometry**:
- The burglar is moving towards the mirror, and the guard can see the burglar in the mirror when the light reflects off the mirror.
- The distance from the mirror to the point directly opposite the burglar is \( \frac{d}{2} = \frac{3}{2} = 1.5 \, m \).

3. **Using the Law of Reflection**:
- According to the law of reflection, the angle of incidence is equal to the angle of reflection.
- When the burglar is at a distance \( x \) from the mirror, the angle formed with the line of sight to the guard can be analyzed.

4. **Setting Up the Triangles**:
- We can create two right triangles:
- Triangle 1: The triangle formed by the burglar, the point directly opposite the burglar, and the mirror.
- Triangle 2: The triangle formed by the guard, the mirror, and the line of sight to the burglar.

5. **Applying Trigonometry**:
- In Triangle 1, we can express \( \cot(\theta) \) as:
\[ \cot(\theta) = \frac{x}{\frac{d}{2}} = \frac{2x}{d} \]
- In Triangle 2, we have:
\[ \cot(\theta) = \frac{d}{d} = 1 \]

6. **Equating the Two Expressions**:
- From the two triangles, we have:
\[ \frac{2x}{d} = 1 \]
- Rearranging gives:
\[ 2x = d \]
- Therefore:
\[ x = \frac{d}{2} \]

7. **Substituting the Value of \( d \)**:
- Given \( d = 3 \, m \):
\[ x = \frac{3}{2} = 1.5 \, m \]

### Conclusion:
The burglar will be \( 1.5 \, m \) away from the mirror when the security guard can first see him in the mirror.