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Products of Reflections



Estimated Time
1 or 2 days
Prerequisites
  • Definitions and fundamental properties of translations, rotations, and reflections.
Materials Needed
  • Copies of the activity pages.
  • Rulers (for drawing line segments and measuring distances).
Ohio Standards Alignment  

Topics

Translations, rotations, and reflections; connections

Overview

Students investigate compositions (called "products") of two reflections and express the result as a single transformation.  There are two cases to consider: 

  • If the reflection lines are parallel, the product is a translation. 
  • If the reflection lines intersect, the product is a rotation.

The "Hook"

Remind students of the various geometric transformations in a plane -- translations, rotations, reflections, and glide reflections.  Reflections can be thought of as "building blocks" out of which all the other transformations can be made. 

Critical Question
Suppose you reflect a geometric figure across one line and then reflect the image across a second line.  Can you accomplish the same result using a single transformation?

The Investigation

Reflect a geometric figure over one line and then reflect its image over a second line that is parallel to the first (see Activity 1).  Compare the final image with the original figure.

Critical Question
What single transformation will give the same result as the two reflections?
Try this for several pairs of parallel reflection lines.
Critical Question
What happens if the order of the reflections is reversed?
Critical Question
What if the two lines are replaced by two other lines parallel to these and the same distance apart?
Next, reflect a geometric figure over one line, and reflect its image over a line that intersects the first line (see Activity 2).  Compare the final image with the original figure.
Critical Question
What single transformation will give the same result as the two reflections?
Try this for several pairs of intersecting reflection lines.  
Critical Question
What happens if the order of the reflections is reversed?
Critical Question
What if two other lines through the same point of intersection are used?
You may want to extend these ideas.
Critical Question
What about a product of three reflections?
Hint:  There are four cases to consider.

Teaching Tips

This activity is done using dot paper.  Several other tools can be used to investigate the same problem. 

After students have solved the problems posed here, they can be challenged to solve the inverse problem:  Given two copies of a geometric shape that are related by a translation or a rotation, find two reflections whose product will yield the same result.  (There are many pairs of reflections that will work in a given situation, but they will all have a common property.)

Synthetic proofs of these two results about products can be based on the defining property of a reflection:  A point and its image are endpoints of a segment whose perpendicular bisector is the reflection line.

A natural extension is to investigate products of three reflections.  Now there are four cases to consider.  If all three reflection lines are parallel or all three are concurrent, the product will be a single reflection.  If two reflection lines are parallel and the third is a tranversal, or if the three lines form a triangle, the product will be a glide reflection.

More complete answers to the questions posed herein can be found here.

Supporting Resources

Citation

From the teaching files of David Kullman.