The angle of incidence of ray is measured from the normal to the plane it intersects and congruent angles have congruent angles have congruent complements, so it follows that the angles of incidence are congruent. Angle AMO and NMB' are vertical angles and vertical angles are congruent. It can be proven triangles MNB and MNB' are congruent by the SAS rule of congruent triangles. Let P be a point also on the mirror on the opposite of M from N but the same distance. Now consider a line dropped from B to B' and call where it intersects the mirror N. With the pencil mark two Xs to indicate the direction of the emergent ray. This line also intersects the mirror at M. This is the incident ray for the angle of incidence, i 10 0. The argument is reversible starting from B and traveling to a reflected point of A, call it A'. So the line from A to B' intersects the mirror at a point M that minimizes A to M to B travel distance/time. The shortest distance between two points is a straight line and by the Pythagorean Theorem it can be shown that the distance form M to B and the distance from M to B' are the same for any point M. One could also consider the problem if a point B' is added as a reflection of the point B. One can use the Pythagorean theorem and calculus to solve this. 100 (1 rating) Transcribed image text: Derive the Law of Reflection, that the angle of incidence equals the angle of reflection, using Fermats Principle. Find a point M on the mirror between these two such that the sum of the distance from A to M and from M to B is minimized. Suppose you have two points A and B above a mirror at different heights above the mirror. These will serve to guide a geometric argument for equal angles. If the speed of light remains constant in the material, the principle of Least Time becomes a principle of least distance. In the case of physical principles, some of these will be just reliable observations.įermat's Least Time Principle tells us that light take the least time path between possible trajectories. A proof requires axioms, basic statements that can't be proven.
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