Problem:
A ray of light originates from point A and travels in a plane, being reflected n times between lines AD and CD, before striking a point B (which may be on AD or CD) perpendicularly and retracing its path to A. If β CDA=8β, what is the largest value n can have?
Answer Choices:
A. 6
B. 10
C. 38
D. 98
E. There is no largest value.
Solution:
Let β DAR1β=ΞΈ and let ΞΈiβ be the (acute) angle the light beam and the reflecting line form at the ith point of reflection. Applying the theorem on exterior angles of triangles to β³AR1βD, then successively to the triangles ΞRiβ1βRiβD,2β©½iβ©½n, and finally to β³RnβBD yields
βΞΈ1β=ΞΈ+8βΞΈ2β=ΞΈ1β+8β=ΞΈ+16βΞΈ3β=ΞΈ2β+8β=ΞΈ+24ββββββββββββΞΈnβ=ΞΈnβ1β+8β=ΞΈ+(8n)β90β=ΞΈnβ+8β=ΞΈ+(8n+8)β.β
But ΞΈ must be positive. Therefore,
0β©½ΞΈnβ=90β(8n+8)β©½882β<11β
If ΞΈ=2β, then n takes its maximum value of 10.
