Answer
The burglar would have to be a distance of $~~1.5~m~~$ from the mirror.
Work Step by Step
The security guard will first see the burglar's image in the right side of the mirror.
A ray between the guard and the left side of the mirror would make an angle of $45^{\circ}$ with the plane of the mirror.
This ray must also make an angle of $45^{\circ}$ from the plane of the mirror to the burglar.
Since the burglar is positioned along the central axis of the mirror, the burglar would have to be a distance of $\frac{d}{2}$ for the guard to see the burglar.
$\frac{d}{2} = \frac{3.0~m}{2} = 1.5~m$
The burglar would have to be a distance of $~~1.5~m~~$ from the mirror.
