5. In a room, the floor is a square with a side length of $6 \mathrm{~m}$, and its center is set as $O$. A lamp $V$ is to be installed on the ceiling directly above $O$. It is known that the angle of illumination of the lamp is $90^{\circ}$ (the maximum angle between the boundaries of the light rays emitted by $V$, i.e., the angle between the boundary of the light rays and $V O$ is $45^{\circ}$). If every part of the room is to be illuminated, the minimum value of $V O$ is $\qquad$ m.
5. $3 \sqrt{2}$.

Let the square $A B C D$ have a side length of $6 \mathrm{~m}$.
When $\triangle V A C$ is an isosceles right triangle, the altitude from $V$ to the hypotenuse $A C$, which is $3 \sqrt{2}$, is the minimum value of $V O$.