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Rigid Transformations: Can you find one?

Author:Ronan Downes, Tibor Marcinek

Part 1 ************************************************************************************

Applets below present a pre-image (blue) and its image (very light blue). Is there a rigid transformation (isometry) from our list that maps the pre-image onto the image? Once you hypothesize which isometry it is, turn it on and adjust the parameters (vectors, angles, centers, lines ...) to demonstrate that it is possible. If impossible, state it so in your answer below.

Case 1

Case 1

1. Did you find such an isometry? What is it?

Select all that apply
  • A
  • B
  • C
  • D
  • E
Check my answer (3)

Case 2

Case 2

2. Did you find such an isometry? What is it?

Select all that apply
  • A
  • B
  • C
  • D
  • E
Check my answer (3)

Case 3

Case 3

3. Did you find such an isometry? What is it?

Select all that apply
  • A
  • B
  • C
  • D
  • E
Check my answer (3)

Case 4

Case 4

4. Did you find such an isometry? What is it?

Select all that apply
  • A
  • B
  • C
  • D
  • E
Check my answer (3)

Your own case

Whether you decide that cases above can or cannot be described using our isometries, try to use the applet below to find a placement of the pre-image and image so that there is no isometry (from our list) that can be used. Start by turning either direct or indirect image on (what's the difference?). Then move it wherever you want and rotate it (by the point) as much as you want. Find a placement of the image so that it could not be described by using isometries we have in our list.

Your playground:

Did you find an arrangement of image and pre-image that shows that our list of isometries is incomplete?

Select all that apply
  • A
  • B
Check my answer (3)

If you answered Yes, please leave the shapes in such position (in the applet above). If you answered No, elaborate more. Do you think there is no such arragement or perhaps there is, it's just you did not find it?

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In your own words, describe the difference between a "direct" and "indirect" image.

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Part 2 ************************************************************************************

In this part you will be asked to find important elements of given transformations. Use the applet at the end to experiment.

Finding Reflection Line

In the applet at the end, turn on Reflection / Flip. If you were told that the blue is the pre-image and yellow is its image under a reflection, how would you find the line of this reflection?

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Finding Rotation Center and Angle

In the applet at the end, turn on Rotation / Turn. If you were told that the blue is the pre-image and yellow is its image under a rotation, how would you find the center of this rotation? How could you find the angle of the rotation?

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Finding Translation Vector

In the applet below, turn on Translation / Slide. If you were told that the blue is the pre-image and yellow is its image under a translation, how would you find its vector?

Check my answer

Playground for Part 2:

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