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GeoGebra

Transformation

Autor:Allan Canonigo

Yout Task

In this GeoGebra applet, you'll explore how translations, reflections, and rotations transform a triangle in the coordinate plane. You begin with Triangle ABC. Your challenge is to reveal other triangles one at a time, identify which transformation(s) map ABC to each one, and discover surprising relationships between different transformations. How to Use the Applet:
  1. Start with only Triangle ABC visible.
  2. Check boxes to reveal triangles one by one: PQR, KLM, VWX, and FGH.
  3. Use checkboxes and sliders to control the transformation tools:
    • x = a and y = b 
    • x = k 
  4. Your goal is to determine how each triangle relates to ABC and compare the triangles to each other.

Check the box for Triangle PQR. Without moving any sliders, compare PQR to ABC.

  • Are they the same size and shape?
  • What stays the same (orientation, angles, side lengths)? What changes (position)?
  • What transformation would map ABC directly to PQR? Explain your reasoning.

Now reveal Triangle KLM. Compare it to ABC.

  • How is KLM oriented relative to ABC? (Check vertex order)
  • Could KLM be created from ABC by a reflection? Why or why not?
  • What transformation (and what specific information) would be needed to map ABC to KLM?

Reveal Triangle VWX. Compare it to ABC.

  • How does the orientation of VWX compare to ABC and KLM?
  • What transformation would map ABC to VWX? How can you tell it's not a translation or rotation?

Using Tools

  • Move the x=k slider. Which triangle(s) match this translation of ABC? What does that tell you?
  • Use the reflection sliders. Which triangle(s) match a reflection? What do you get when you reflect twice?
  • Use the rotation slider (if available). Which triangle matches a rotation? What rotation angle matches a 180° turn?

Comparing Transformations

  • Compare KLM and VWX. Can a reflection equal a rotation? If yes, what must be true?
  • Reveal FGH. Can you create FGH from ABC with one transformation? With two? Try two reflections.

Equivalences

  • Reflect ABC over x=0 then y=0. What single transformation results? How does the angle between lines relate?
  • Reflect ABC over x=1 then x=3. What single transformation occurs? How does the line distance relate?
  • What do these equivalences imply about reflections, rotations, and translations? Find two different ways to map ABC to FGH.

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