Minimum Perimeter

The shortest length or distance of $LMN$ will be the complete path traveled by light. Treating the internal sides of $\triangle ABC$ as mirrors, this occurs when a beam of light starting from $M$ , hitting the mid-point of $AB$ at an incident angle of $30^circ$ leaving with a reflective angle of $30^circ$ ; again hitting and leaving the mid-point of $BC$ at $30^circ$ before returning to point $M$ . Therefore, $L$ and $N$ are mid-points of $AB$ and $BC$ . and $\triangle LMN$ is an equilateral triangle with side length of $4$ cm and hence its perimeter $4\times 3 = \boxed{12}$ .