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Compositions of Reflections Theorems

Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

Reflections are one type of mathematical transformation. This lesson describes reflections and the other three main transformations. We'll look at compositions of reflections over various sets of lines and how they connect to single transformations.

Transformations

When you were a kid, did you ever put a sticker on your bicycle tire so that everyone could see it go round and round as you rode your bike? If so, you probably didn't realize it, but you did something mathematical! You see, the sticker rotating around the center of the tire is called a rotation in mathematics, and it is a type of transformation.


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A transformation of an object moves or resizes the object in a specific way. There are four main types of transformations:

  1. Rotations involve turning an object around a point.
  2. Reflections involve flipping an object over a line.
  3. Translations involve sliding an object.
  4. Resizing involves making an object larger or smaller by some factor.

These are all pretty straightforward. Let's take it a step further with compositions of transformations. Compositions of transformations involve performing a transformation on an object and then performing another transformation on the result. For example, consider translating a rectangle four units to the right and then rotating it clockwise 90 degrees around a point.


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This is an example of a composition of transformations, because we first translated the rectangle, then we rotated the result around a point.

Where it get's really neat is in the fact that certain compositions of transformations are equivalent to a single transformation. Three specific ones are described in compositions of reflections theorems. Let's discuss!

Compositions of Reflections in Parallel Lines

The compositions of reflections over parallel lines theorem states two things:

  1. If we perform a composition of two reflections over two parallel lines, the result is equivalent to a single translation transformation of the original object.
  2. If we perform a composition of three reflections over three parallel lines, the result is equivalent to a single reflection transformation of the original object.

To illustrate the first part of this theorem, let's perform a composition of reflections on a triangle over two parallel lines. In other words, let's reflect the triangle over one of the lines and then reflect the resulting image over the other line.


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See how reflecting the triangle over the line x = 4 and then reflecting the result over the parallel line x = 8 is the same as if we had just translated (slid) the original triangle eight units to the right? This illustrates the first part of the theorem.

The second part of the theorem has to do with a composition of three reflections. Suppose we took the result from our earlier composition and reflected it over another parallel line, x = 12.


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In doing this, we've reflected the original triangle over three parallel lines, and we can see that this is equivalent to reflecting the original triangle just once over the line x = 8. This illustrates the second part of the theorem.

Isn't this neat? Let's consider another theorem about compositions of transformations.

Compositions of Reflections in Intersecting Lines

The compositions of reflections over intersecting lines theorem states that if we perform a composition of two reflections over two lines that intersect, the result is equivalent to a single rotation transformation of the original object.

To illustrate this, let's give it a try! Let's reflect a triangle over two intersecting lines, specifically the x and y axes, so we can see that the result is equivalent to a single rotation transformation of the original triangle.


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