In this GeoGebra applet, you'll explore how [b]translations[/b], [b]reflections[/b], and [b]rotations[/b] transform a triangle in the coordinate plane. You begin with [b]Triangle ABC[/b]. 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.[br][br][b]How to Use the Applet:[/b][list=1][*]Start with only Triangle ABC visible.[/*][*]Check boxes to reveal triangles one by one: [b]PQR[/b], [b]KLM[/b], [b]VWX[/b], and [b]FGH[/b].[/*][*]Use checkboxes and sliders to control the transformation tools:[list][*][b]x = a[/b] and [b]y = b[/b] [/*][*][b]x = k[/b] [/*][/list][/*][*]Your goal is to determine how each triangle relates to ABC and compare the triangles to each other.[/*][/list]
Check the box for [b]Triangle PQR[/b]. Without moving any sliders, compare PQR to ABC.[list][*]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 reas[color=#f9fafb]oning.[/color][/*][/list]
Now reveal [b]Triangle KLM[/b]. Compare it to ABC.[list][*]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 [color=#f9fafb]KLM?[/color][/*][/list]
Reveal [b]Triangle VWX[/b]. Compare it to ABC.[list][*]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?[/*][/list]
[b]Using Tools[/b][br][list][*]Move the [b]x=k[/b] slider. Which triangle(s) match this translation of ABC? What does that tell you?[br][/*][*]Use the reflection sliders. Which triangle(s) match a reflection? What do you get when you reflect twice?[br][/*][*]Use the rotation slider (if available). Which triangle matches a rotation? What rotation angle matches a 180° turn?[br][/*][/list]
[b]Comparing Transformations[/b][br][list][*]Compare KLM and VWX. Can a reflection equal a rotation? If yes, what must be true?[br][/*][*]Reveal FGH. Can you create FGH from ABC with one transformation? With two? Try two reflections.[br][/*][/list]
[b][color=#20124d]Equivalences[/color][/b][br][list][*]Reflect ABC over x=0 then y=0. What single transformation results? How does the angle between lines relate?[br][/*][*]Reflect ABC over x=1 then x=3. What single transformation occurs? How does the line distance relate?[br][/*][*]What do these equivalences imply about reflections, rotations, and translations? Find two different ways to map ABC to FGH.[br][/*][/list]