2021 AMC 12A Problems/Problem 11
Contents
Problem
A laser is placed at the point 
. The laser bean travels in a straight line. Larry wants the beam to hit and bounce off the 
-axis, then hit and bounce off the 
-axis, then hit the point 
. What is the total distance the beam will travel along this path?
Diagram
~MRENTHUSIASM (by Desmos: https://www.desmos.com/calculator/y6wyqok8gm)
Solution 1
Every time the laser bounces off a wall, instead we can imagine it going straight by reflecting it about the wall. Thus, the laser starts at 
and ends at 
, so the path's length is 
~JHawk0224
Solution 2 (Detailed Explanation of Solution 1)
Let 
be the point where the beam hits the -axis, and 
be the point where the beam hits the -axis.
Reflecting 
about the -axis gives 
Then, reflecting 
over the -axis gives 
Finally, reflecting 
about the -axis gives 
as shown below.
It follows that 
The total distance that the beam will travel is
Graph in Desmos: https://www.desmos.com/calculator/lqatf2r9bu
~MRENTHUSIASM
Solution 3 (Slopes and Parallelogram)
Define points 
and 
as Solution 2 does.
~MRENTHUSIASM
Video Solution by OmegaLearn (Using Reflections and Distance Formula)
~ pi_is_3.14
Video Solution by Hawk Math
https://www.youtube.com/watch?v=AjQARBvdZ20
See also
| 2021 AMC 12A (Problems • Answer Key • Resources) | |
| Preceded by Problem 10 |
Followed by Problem 12 |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
| All AMC 12 Problems and Solutions | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
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