Difference between revisions of "1981 AHSME Problems/Problem 20"
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A ray of light originates from point <math>A</math> and travels in a plane, being reflected <math>n</math> times between lines <math>AD</math> and <math>CD</math> before striking a point <math>B</math> (which may be on <math>AD</math> or <math>CD</math>) perpendicularly and retracing its path back to <math>A</math> (At each point of reflection the light makes two equal angles as indicated in the adjoining figure. The figure shows the light path for <math>n=3</math>). If <math>\measuredangle CDA=8^\circ</math>, what is the largest value <math>n</math> can have? | A ray of light originates from point <math>A</math> and travels in a plane, being reflected <math>n</math> times between lines <math>AD</math> and <math>CD</math> before striking a point <math>B</math> (which may be on <math>AD</math> or <math>CD</math>) perpendicularly and retracing its path back to <math>A</math> (At each point of reflection the light makes two equal angles as indicated in the adjoining figure. The figure shows the light path for <math>n=3</math>). If <math>\measuredangle CDA=8^\circ</math>, what is the largest value <math>n</math> can have? | ||
| − | + | <asy> | |
unitsize(1.5cm); | unitsize(1.5cm); | ||
pair D=origin, A=(-6,0), C=6*dir(160), E=3.2*dir(160), F=(-2.1,0), G=1.5*dir(160), B=(-1.4095,0); | pair D=origin, A=(-6,0), C=6*dir(160), E=3.2*dir(160), F=(-2.1,0), G=1.5*dir(160), B=(-1.4095,0); | ||
| Line 32: | Line 32: | ||
draw(anglemark(B,G,D,12)); | draw(anglemark(B,G,D,12)); | ||
draw(anglemark(B,G,D,16)); | draw(anglemark(B,G,D,16)); | ||
| − | </asy | + | </asy> |
| − | < | + | <math> \textbf{(A)}\ 6\qquad\textbf{(B)}\ 10\qquad\textbf{(C)}\ 38\qquad\textbf{(D)}\ 98\qquad\textbf{(E)}\ \text{There is no largest value.} </math> |
Revision as of 18:22, 23 October 2021
Problem
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?
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?
![[asy] unitsize(1.5cm); pair D=origin, A=(-6,0), C=6*dir(160), E=3.2*dir(160), F=(-2.1,0), G=1.5*dir(160), B=(-1.4095,0); draw((-6.5,0)--D--C,black); draw(A--E--F--G--B,black); dotfactor=4; dot("$A$",A,S); dot("$C$",C,N); dot("$R_1$",E,N); dot("$R_2$",F,S); dot("$R_3$",G,N); dot("$B$",B,S); markscalefactor=0.015; draw(rightanglemark(G,B,D)); draw(anglemark(C,E,A,12)); draw(anglemark(F,E,G,12)); draw(anglemark(E,F,A)); draw(anglemark(E,F,A,12)); draw(anglemark(B,F,G)); draw(anglemark(B,F,G,12)); draw(anglemark(E,G,F)); draw(anglemark(E,G,F,12)); draw(anglemark(E,G,F,16)); draw(anglemark(B,G,D)); draw(anglemark(B,G,D,12)); draw(anglemark(B,G,D,16)); [/asy]](http://latex.artofproblemsolving.com/5/e/d/5ed239b250fc1f8c6a259db32194b749fea66099.png)
