Today we will investigate an isometry called a glide reflection. While glide reflection is not a tranformation found in the Transform menu, you will define it as a custom transformation, and in the process, learn what a glide reflection is and what it does.
Sketch 1
Step 1 : Construct a polygon interior whose orientation will be easy to keep track of. (In other words, make it scalene with no lines of symmetry.)
Step 2 : Construct a line EF and a point G on the line so that E and G are about an inch apart.
Step 3 : Select E and G and choose Mark Vector in the Transform menu.
Step 4 : Mark line EF as mirror in the Transform menu.
Step 5 : Reflect your figure about line EF.
Step 6 : Translate this image by the marked vector EG. This second image is a glide reflection of your original figure. Hide the intermediate image (the first reflection).
Step 7 : Select the glide reflected image and the original image and choose Define Transform. Define the transform as "Glide Reflection".
Step 8 : Use Glide Reflection to construct several more images as shown.
Investigation 1
Manipulate your original figure and watch the effect on the images. A glide reflection is the product of what two transformations? A translation is the product of what two transformations? So a glide reflection can be thought of as the product of what three transformations?
Sketch 2
Step 1 : Construct a polygon interior whose orientation will be easy to keep track of. (In other words, make it scalene with no lines of symmetry.)
Step 2 : Construct a line EF and a point G on the line so that E and G are about an inch apart.
Step 3 : Select E and G and choose Mark Vector in the Transform menu.
Step 4 : Mark line EF as mirror in the Transform menu.
Step 5 : Translate your pre-image by the marked vector EG.
Step 6 : Reflect your image about line EF.
Investigation 2
Is this new sketch a glide reflection? Did it give the same result as did sketch 1? Would all reflections followed by a translation result in the same image as the same translation and then the reflection?
Assignment - Day 6
1. Answer the questions posed in the investigations above.
2. Describe the new ideas you learned about today.
3. Discuss any questions or points of misunderstanding you may have had in the course of this lesson.
4. Create another example of your own glide reflection. Be sure to show the line of reflection as well as the vector of translation.
5. Create an example of a reflection followed by a translation that does NOT produce a glide reflection.
Resources: Some of this material is taken from Exploring Geometry with The Geometer's Sketchpad, Blackline Activity Masters for Use with The Geometer's Sketchpad. Key Curriculum Press. 1996.