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### Light Beam Reflection

```
Date: 01/31/2002 at 07:02:08
From: Teri Lurie
Subject: Algebra

Please can you help me with this question?

Four mirrors form a rectangle 3 m by 2 m. A light beam is shone from A
at 45 degrees. Which corner does the beam strike first?

Teri
```

```
Date: 01/31/2002 at 12:17:13
From: Doctor Peterson
Subject: Re: Algebra

Hi, Teri.

You can either do this in a very simple way, by just drawing what
happens, or in a way that demands more insight but makes the problem a
lot more interesting.

First, you can just draw it out:

B             C
+---+---o---+
|   | / |   |
+---/---+---+
| / |   |   |
o---+---+---+
A             D

As I've shown, the beam will first hit the wall 2/3 of the way along
side BC, at coordinates (2,2) if A is the origin. Then it will reflect
at the same 45 degree angle back down toward side CD. Draw where it
will hit that:

B             C
+---+---o---+
|   | / | \ |
+---/---+---o
| / |   |   |
o---+---+---+
A             D

(It will be very useful to realize that a 45 degree angle will always
take the beam one unit in the y direction for each unit it goes in the
x direction.)

Keep on like this, and you will soon find the answer.

Now for some deeper insight:

Instead of thinking of the beam reflecting, you can think of the
reflections of the "pool table" itself (that's how I think of it) in
the mirrors:

A---+---+---D---+---+---A
|   |   |   |   |   |   |
+---+---+---o---+---+---+
|   |   | / |   |   |   |
B---+---o---C---+---+---B
|   | / |   |   |   |   |
+---/---+---+---+---+---+
| / |   |   |   |   |   |
A---+---+---D---+---+---A

If you were standing on the "table," you would see many copies of the
table (and yourself) over and over. And the path of a light beam would
seem to be a straight line cutting through many of these reflected
copies. I've shown its path from A to side BC and on to a reflected
copy of side CD, matching what I showed before. Keep going, and it
will eventually hit a reflection of a corner, and that will be your

hits will be at coordinates (x,y) where x and y are equal (so that it
is on the path of the light beam), but x is a multiple of 3 and y is a
multiple of 2 (so that it is a reflection of a corner). Can you see
how to use this fact to solve the problem without having to follow the
beam all the way? That would be very useful for solving the problem
more generally, for different sized tables and different directions.

You can find some related problems in our archive at

Bouncing Cue Ball
http://mathforum.org/dr.math/problems/nichole10.29.96.html

Pool Table Algebra
http://mathforum.org/dr.math/problems/roland10.21.98.html

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Coordinate Plane Geometry
High School Euclidean/Plane Geometry
High School Geometry

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