Difference between revisions of "2021 AMC 12A Problems/Problem 11"
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==Solution 2 (Detailed Explanation of Solution 1)== | ==Solution 2 (Detailed Explanation of Solution 1)== | ||
− | Let <math>A=(3,5), D=(7,5), B</math> be the point where the beam | + | Let <math>A=(3,5), D=(7,5), B</math> be the point where the beam hits the <math>y</math>-axis, and <math>C</math> be the point where the beam hits the <math>x</math>-axis. |
Reflecting <math>\overline{BC}</math> about the <math>y</math>-axis gives <math>\overline{BC'}.</math> Then, reflecting <math>\overline{CD}</math> over the <math>y</math>-axis gives <math>\overline{C'D'}.</math> Finally, reflecting <math>\overline{C'D'}</math> about the <math>x</math>-axis gives <math>\overline{C'D''},</math> as shown below. | Reflecting <math>\overline{BC}</math> about the <math>y</math>-axis gives <math>\overline{BC'}.</math> Then, reflecting <math>\overline{CD}</math> over the <math>y</math>-axis gives <math>\overline{C'D'}.</math> Finally, reflecting <math>\overline{C'D'}</math> about the <math>x</math>-axis gives <math>\overline{C'D''},</math> as shown below. |
Revision as of 19:37, 14 February 2021
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.