1981 AHSME Problems/Problem 20
A ray of light originates from point and travels in a plane, being reflected times between lines and before striking a point (which may be on or ) perpendicularly and retracing its path back to (At each point of reflection the light makes two equal angles as indicated in the adjoining figure. The figure shows the light path for ). If , what is the largest value can have?
Notice that when we start, we want the smallest angle possible of reflection. The ideal reflection would be , but that would be impossible. Therefore we start by working backwards. Since angle is , the reflection would give us a triangle with angles , and . Then, when we reflect again, we will have = . Since the other side of the reflection when we had the degrees had carried over to the other side, we have a triangle.
Notice that we keep decreasing by increments of . This is because the starting angle was and since we always have to decrease every time and that every triangle has every increasing angles of , we must decrease by every time. This is the most optimal path of the light beam.
The pattern of light will be . When we get to the angle of degrees, we have reached angle . Therefore, we don't count the , so our total number of reflections between and is