This worksheet helps build visual intuition for the associativity of the composition of reflections. Before working on, make sure you have a good understanding of (see our previous activities):[br][br]1. How translation and rotation can result from the composition of two line reflections.[br]2. How the translation vector or rotation angle/center relate to the position of these two lines of reflection.
We want to demonstrate that [br][size=150][center][b](a [/b][math]\bigcirc[/math][b] b) [/b][math]\bigcirc[/math][b] c = a [/b][math]\bigcirc[/math][b] (b [/b][math]\bigcirc[/math][b] c)[/b] . [/center][size=100]To proceed, we need to interpret what these expressions mean for the composition of line reflections. [i]Answer the questions that follow based on the picture in the applet[/i] (you may need to scroll down to see the applet).[/size][/size]
Left side:[br][br]a [math]\bigcirc[/math] (b [math]\bigcirc[/math] c) is a reflection about the line [i]a[/i] followed by (b [math]\bigcirc[/math] c). What is (b [math]\bigcirc[/math] c)? Can you replace it with a single isometry? Describe it![br][br]Once you interpreted the left side, apply it to the given triangle: Reflect it about the line [i]a[/i] and follow by the isometry you described above.[br][br][i]Format the intermidiate and final images (color, transparency) to distinguish them clearly. Label the final image LS (left side).[br][br][i]If you need to type greek letters, press Alt (or Ctrl on Macs) + keyboard letter. For example, Alt + b will insert [math]\beta[/math] (on a PC).[/i][/i]
Two reflections with intersecting lines will produce a rotation. Make sure you know where the center of such a rotation is and what its angle is.[br][br]Two reflections with parallel lines will produce a translation. Make sure you know the position and lenght of the translation vector.
Right side:[br][br](a [math]\bigcirc[/math] b) [math]\bigcirc[/math] c is composition (a [math]\bigcirc[/math] b) followed by a reflection about the line [i]c[/i] . What is (a [math]\bigcirc[/math] b)? Can you replace it with a single isometry? Describe it![br][br]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 [i]c.[/i][br][br][i]Format the intermidiate and final images (color, transparency) to distinguish them clearly. Label the final image RS (right side).[br][br][i][i]If you need to type greek letters, press Alt (or Ctrl on Macs) + keyboard letter. For example, Alt + b will insert [math]\beta[/math][/i][/i][/i] (on a PC).
Two reflections with intersecting lines will produce a rotation. Make sure you know where the center of such a rotation is and what its angle is.[br][br]Two reflections with parallel lines will produce a translation. Make sure you know the position and lenght of the translation vector.
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? [br][br]2. Is this a proof of associative property for line reflections?
2. It's better to call it "convincing demonstration" than a proof. Why?