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Two reflections - three explorations

Autor:Tibor Marcinek
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 cannot be a result of the composition of two line reflections? Explain.

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Case 1

1a. Play with the two lines in the applet below. What do you notice about them? 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. 1c. What do you notice about the pre-image and the final image? Which isometry was generated by the two lines of reflection? 1.d To verify your observation, create the isometry from 1c using GeoGebra tools* and try to match the two images. * 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. 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? * 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?  

Case 1

Case 1

1a. What did you notice about the lines?

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Case 1

1b and 1e. What is the resulting isometry and how does it relate to the two given lines?* * 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.

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Case 2

1a. Play with the two lines in the applet below. What do you notice about them? 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. 1c. What do you notice about the pre-image and the final image? Which isometry was generated by the two lines of reflection? 1.d To verify your observation, create the isometry from 1c using GeoGebra tools* and try to match the two images. * 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. 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? * 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?  

Case 2

Case 2

1a. What did you notice about the lines?

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Case 2

1b and 1e. What is the resulting isometry and how does it relate to the two given lines?* * 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.

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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?

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If you want, in the applet below, you may draw such an arrangement and test it.

Case 3

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