1. Construct an equilateral triangle that stays equilateral no matter how you move the points around. (To do this, start with a segment and use the construction that Euclid describes in his Prop. 1.) Likewise, construct a square that remains a square. (You'll need to think about how to do this. You'll want to start with another segment, and construct a square on the segment. One useful command for this is "Construct Perpendicular Line," which is in the construct menu. ) Save your sketch as “sym1”.
2. For each of these, see how many symmetries you can find. Construct a new arbitrary line, and reflect the figures over it. To do this, select the line, then go to "Mark Mirror" in the Transform menu. Then, select the things you want to reflect, go to the Transform menu, and choose "reflect." After you have reflected the figures over the line, you can move the line around, and see what reflection symmetries you can find. When you find one, the figure will coincide with its reflection. Also, create a new point and a new angle (made from two line segments), and rotate the figures about the point by the angle, using the transform menu. Again, see if you can find symmetries by making the figure coincide with its rotation.
3. Now go back to your saved sketch sym1. For each symmetry that you found, construct the line/point of symmetry directly from the original figure, and reflect/rotate the figure about that line/point. If you’ve constructed it correctly, the image should stay on top of the original figure no matter how you move it.
4. Finally, for each symmetry that you found, try to see what kinds of triangles or quadrilaterals have that symmetry. You can do this by starting with a line or point of symmetry and a point or segment of a figure, reflecting/rotating it about the point or line, and seeing how much of the figure is determined. What happens if you require two or more symmetries?
Homework: Write up a lab report explaining your results. You can do this by yourself or in pairs. Your report should include: 1) A description of each of the symmetries that you found, including explicit instructions for how to find/construct the points, lines, and angles of symmetry; and 2) For each set of symmetries on your list, a complete list of all of the types of triangles and quadrilaterals that have that symmetry.