Congruence Transformations
Collection of worksheets to discover important theorems regarding the composition of reflections.
Table of Contents
- Finding isometries - Tools
- Finding Images on a square grid
- Finding images with a transparency
- [A] Finding images with a transparency
- Finding Isometries - Cases
- Isometries - Case1
- Isometries - Case2
- Isometries - Case3
- Isometries - Case 4 (independent R&T)
- Isometries - Case4 (forced parallel)
- Isometries - Can you find one?
- More practice on Finding Isometries (intuitive approach)
- Properties of isometries
- Composition of Isometries - is it commutative?
- Is composition of reflections associative?
- Is our list complete?
- What is a Turn-Slide?
- What is a Turn-Turn?
- What is a Turn-Flip?
- Is our list complete? (Not animated)
- What is a Turn-Turn? (Static)
- What is a Turn-Flip ? (no pivoting)
- What is a Turn-Slide? (Static)
- Composition of reflections
- Two reflections - three explorations
- Composition of 3 reflections - Concurrent lines
- Composition of 3 reflections - Parallel lines
- Composition of 3 reflections - General position
Finding Images on a square grid
Find the images of the triangle DCE under the following transformations:[br][br]1. Translation given by the vector u.[br]2. Line reflection given by the line a.[br]3. Rotation 90° clockwise about the point H[br]4. Glide reflection given by the line a and vector u.[br][br]Explain how you use the square grid to find to find the images. You may print the applet image (applet menu > Print preview) to work on it.[br][br]Make sure that you know how find images also by using [url=https://www.geogebra.org/m/pshx6vh9]a transparency[/url] and dynamic geometry (using the tools provided in the applet).
Isometries - Case1
Composition of Isometries - is it commutative?
In previous activities, you played with isometries that were created for you. To answer questions in this activity, you will have to use GeoGebra tools to create your own isometries. Use your constructions to support your answers.[br][br][i]If you need to type greek letters, pres Alt (or Ctrl on Macs) + keyboard letters. For example, Alt + b will insert [/i][math]\beta[/math][i] (on a PC).[/i]
1. Is "Turn slide" the same as "Slide turn"?
In other words, is composition of translation and rotation commutative?[br][br]1a. In the applet below, provide a clear visual argument to support your answer. Use a slider to define a rotation.[br][br]1b. Is there a special case (certain specific translations or rotations), for which the composition would be commutative?
1. Is "Turn slide" the same as "Slide turn"?
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2. Is "Slide flip" the same as "Flip slide"?
In other words, is composition of translation and reflection commutative?[br][br]2a. In the applet below, provide a clear visual argument to support your answer. [br][br]2b. Is there a special case (certain specific translations or reflections), for which the composition would be commutative?[br][br]2c. Use your answer in 2b to discuss the relationship between a "Slide flip" and Glide Reflection.
2. Is "Slide flip" the same as "Flip slide"?
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3. Is "Turn flip" the same as "Flip turn"?
In other words, is composition of reflection and rotation commutative?[br][br]3a. In the applet below, provide a clear visual argument to support your answer. Use a slider to define a rotation.[br][br]3b. Is there a special case (certain specific reflections or rotations), for which the composition would be commutative?
3. Is "Turn flip" the same as "Flip turn"?
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4. Is composition of two line reflections commutative?
4a. In the applet below, provide a clear visual argument to support your answer.[br][br]4b. Is there a special case, for which the composition would be commutative?
4. Is composition of two line reflections commutative?
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5. Are there isometries, the composition of which is always commutative?
5a. In the applet below, provide a clear visual argument to support your answer. If needed, use a slider to define a rotation.
5. Are there isometries, the composition of which is always commutative?
What is a Turn-Slide?
What is a composition of rotation and translation? What can it be?[br][br]Use decomposition into reflections to reason about the exact characteristics of the resulting isometry.
What is a Turn-Turn? (Static)
What is a composition of two rotations? What can it be?
Two reflections - three explorations
In this activity, you will be exploring what will happen when we compose two line reflections. Obviously, there are two cases to explore: when the two lines are parallel and when they are not.
Preliminaries
0. Which isometries [u]cannot[/u] be a result of the composition of two line reflections? Explain.
Case 1
1a. Play with the two lines in the applet below. What do you notice about them?[br][br]1b. Reflect the given blue polygon (pre-image) about one of the lines. Then reflect the obtained image about the other line. Format the images (color, transparency) to make obvious obvious which is the pre-image, the final image and intermediary image.[br][br]1c. What do you notice about the pre-image and the final image? Which isometry was generated by the two lines of reflection?[br][br]1.d To verify your observation, create the isometry from 1c using GeoGebra tools* and try to match the two images.[br][size=85]* If it is a translation, construct a vector; if it is a rotation, place its center and create a slider for angle, if it is a glide reflection, place a line and vector. Then use GeoGebra's transformations to create a new image. Format the new image to make it stand out.[br][/size][br]1e. When you have matched the two images in 1d., make a more detailed observation: How does the resulting isometry relate to the two given lines?* What happens when you play with the lines?[br][size=85]* If it is a translation, how the direction and length of the vector relate to the two lines? If it is rotation, how does the center and angle relate to the two lines? If it is a glide reflection or reflection, where is the line and vector in relation to the two lines? [/size]
Case 1
Case 1
1a. What did you notice about the lines?
Case 1
1b and 1e. What is the resulting isometry and how does it relate to the two given lines?*[br][size=85]* For a translation, explain how the direction and length of the vector relate to the two lines. For rotation, explain how the center and angle relate to the two lines. For a glide reflection explain the position of the line and vector in relation to the two lines.[/size]
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Case 2
1a. Play with the two lines in the applet below. What do you notice about them?[br][br]1b. Reflect the given blue polygon (pre-image) about one of the lines. Then reflect the obtained image about the other line. Format the images (color, transparency) to make obvious obvious which is the pre-image, the final image and intermediary image.[br][br]1c. What do you notice about the pre-image and the final image? Which isometry was generated by the two lines of reflection?[br][br]1.d To verify your observation, create the isometry from 1c using GeoGebra tools* and try to match the two images.[br][size=85]* If it is a translation, construct a vector; if it is a rotation, place its center and create a slider for angle, if it is a glide reflection, place a line and vector. Then use GeoGebra's transformations to create a new image. Format the new image to make it stand out.[br][/size][br]1e. When you have matched the two images in 1d., make a more detailed observation: How does the resulting isometry relate to the two given lines?* What happens when you play with the lines?[br][size=85]* If it is a translation, how the direction and length of the vector relate to the two lines? If it is rotation, how does the center and angle relate to the two lines? If it is a glide reflection or reflection, where is the line and vector in relation to the two lines? [/size]
Case 2
Case 2
1a. What did you notice about the lines?
Case 2
1b and 1e. What is the resulting isometry and how does it relate to the two given lines?*[br][size=85]* For a translation, explain how the direction and length of the vector relate to the two lines. For rotation, explain how the center and angle relate to the two lines. For a glide reflection explain the position of the line and vector in relation to the two lines.[/size]
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Case 3
3a. You probably concluded that two lines of reflection cannot generate a glide reflection. Based on your observations and discoveries above, can you suggest an arrangement of three lines that will generate a glide reflection?
If you want, in the applet below, you may draw such an arrangement and test it.