Is composition of reflections associative?
Prelimiaries
Associative Property
(a b) c = a (b c) .
To proceed, we need to interpret what these expressions mean for the composition of line reflections. Answer the questions that follow based on the picture in the applet (you may need to scroll down to see the applet).Left side: a (b c) is a reflection about the line a followed by (b c). What is (b c)? Can you replace it with a single isometry? Describe it! Once you interpreted the left side, apply it to the given triangle: Reflect it about the line a and follow by the isometry you described above. Format the intermidiate and final images (color, transparency) to distinguish them clearly. Label the final image LS (left side). If you need to type greek letters, press Alt (or Ctrl on Macs) + keyboard letter. For example, Alt + b will insert (on a PC).
Right side: (a b) c is composition (a b) followed by a reflection about the line c . What is (a b)? Can you replace it with a single isometry? Describe it! Once you interpreted the right side, apply it to the given triangle: Start with the isometry you described above followed by a reflection about the line c. Format the intermidiate and final images (color, transparency) to distinguish them clearly. Label the final image RS (right side). If you need to type greek letters, press Alt (or Ctrl on Macs) + keyboard letter. For example, Alt + b will insert (on a PC).
Conclusion
1. Observe the final images for the left and right sides. Turn on the lines' control points and use them to change the position of the lines. What's happening with the LS and RS images? What does it demonstrate? 2. Is this a proof of associative property for line reflections?