2021 AMC 12A Problems/Problem 11
Contents
Problem
A laser is placed at the point . The laser beam 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
Solution 1 (Geometry)
Let and
Suppose that the beam hits and bounces off the
-axis at
then hits and bounces off the
-axis at
When the beam hits and bounces off a coordinate axis, the angle of incidence and the angle of reflection are congruent. So, we straighten up the path of the beam by reflections:
- We reflect
about the
-axis to get
- We reflect
about the
-axis to get
- We reflect
about the
-axis to get
We obtain the following diagram:
It follows that
and
The total distance that the beam will travel is
~MRENTHUSIASM (Solution)
~JHawk0224 (Proposal)
Solution 2 (Algebra)
Define points and
as Solution 1 does.
When a straight line hits and bounces off a coordinate axis at point the ray entering
and the ray leaving
have negative slopes. Let
be the line containing
and perpendicular to that coordinate axis. Geometrically, these two rays coincide when reflected about
Let the slope of be
It follows that the slope of
is
and the slope of
is
Here, we conclude that
Next, we locate on
such that
We obtain parallelogram
as shown below.
Let
In parallelogram
we get
By symmetry, we get
Applying the slope formula to and
gives
Equating the last two expressions gives
By the Distance Formula, we have and
The total distance that the beam will travel is
~MRENTHUSIASM
Solution 3 (Answer Choices and Educated Guesses)
Define points and
as Solution 1 does.
Since choices and
all involve
we suspect that one of them is the correct answer. We take a guess in faith that
and
all form
angles with the coordinate axes, from which
and
The given condition
verifies our guess, as shown below.
Following the last paragraph of Solution 2 gives the answer
~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
Video Solution by TheBeautyofMath
~IceMatrix
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 |
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