2021 AMC 12A Problems/Problem 11

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

File:2021 AMC 12A Problem 11.png

~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.

File:2021 AMC 12A Problem 11(2).png

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.

When a line segment hits and bounces off a coordinate axis at point the ray entering and the ray leaving have negative slopes. Geometrically, the rays coincide when reflected about the line perpendicular to that coordinate axis, creating a line symmetry. 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 thus is a parallelogram, as shown below.

File:2021 AMC 12A Problem 11(3).png

Let By the property of slopes, we get By symmetry, we obtain

Applying the slope formula on and gives Equating the last two expressions gives

By the Distance Formula, and The total distance that the beam will travel is

Graph in Desmos: https://www.desmos.com/calculator/cp9uauhl6k

~MRENTHUSIASM

Solution 4 (Answer Choices and Educated Guesses)

Define points and as Solution 2 does.

Since choices and all involve we suspect that one of them is the correct answer. We take a guess in faith that and form angles with the coordinate axes, then we get that and This result verifies our guess. Following the penultimate paragraph of Solution 3 gives the answer

~MRENTHUSIASM

Video Solution by OmegaLearn (Using Reflections and Distance Formula)

https://youtu.be/e7tNtd-fgeo

~ pi_is_3.14

Video Solution by Hawk Math

https://www.youtube.com/watch?v=AjQARBvdZ20

Video Solution by TheBeautyofMath

https://youtu.be/ySWSHyY9TwI

~IceMatrix

See also

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.