The mirror dome

A ray of light is emitted from a point on the wall of a mirror-covered semicircle and reflects four times as shown to return to the same spot. If the radius is 1, what is the length of the path?

Scroll down for a solution to this problem.


The path length is 3√3.

The crucial step is to use the mirror trick every time the ray reflects in the diameter. The semicircle basically becomes equivalent to the circle, and for the circle it is straightforward that the path must be an equilateral triangle, which has perimeter 3√3 times the radius.

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