Explore these with Sketchpad. Suggestions will be made in lab.
Line reflection products
Let M, N, P be reflections in lines through O.
What isometry is MN? What isometry is NM? What is (MN)(NM)? What is (NM)(MN)? What is (MN)(MN)?
In what cases is it true that MN = NM? (Important!)
Construct a line reflection Q so that PQ = MN. Construct a line reflection R so that RP = MN.
Perpendicular lines
Let K, L, M be line reflections with mirror lines k, l, m. Suppose that k is perpendicular to l and m. What isometry is KLM? How is it related to LKM and LMK? Use this to answer the question of what is HN, if H is a point reflection and N is a line reflection?
Products of rotations.
Construct a sketch illustrating how to construct the center of a product of rotations (which is a rotation).
Other Products
Product of two glide reflections
Product of a translation and a line reflection or a line reflection and a translation.
Product of a translation and a rotation or a rotation and a translation.
Product of a rotation with center A and angle t with a rotation with center B and angle –t.