Among the possible questions would be questions of the following sort:
Given an isometry X of a given type and an isometry Y of a given type, what kind of isometry is XY? This is a short answer question, since no computation or detailed work is required to answer.
Example 1. If X is a point reflection and Y is a line reflection, what kind of isometry is XY?
Answer: XY is a glide reflection.
Example 2. If X is a rotation and Y is a rotation, what kind of isometry is XY?
Answer: XY is either a rotation or a translation.
Note: Since X or Y can be a line reflection, rotation, point reflection (special case of rotation), translation, or glide reflection, there are 5 possible choices for X and 5 for Y. You should be able to answer confidently for each case.
Given an explicit isometry X and an explicit isometry Y, what explicit isometry is XY? This is a longer answer question, since a computation or detailed work is required to answer.
Example 3A. You are given a figure with points A and line b. If X is a point reflection in A and Y is a line reflection in b, tell what type of isometry is XY and construct the defining data of XY.
Answer: XY is a glide reflection. The invariant line in this case is the line g through A perpendicular to b, so you would construct this line g. The vector for the translation is vector A'A, where A' is the reflection of A in b, so you will construct A' also.
Reason: Let a be the line through A perpendicular to g. Then X = RgRa, so XY = Rg(RaRb). But RaRb is the translation that takes A' to A. Notice that vector A'A is parallel to g as required in the definition of glide reflection.
Example 3B. You are given a figure with points A and line b. If X is a point reflection in A and Y is a line reflection in b, tell what type of isometry is YX and construct the defining data of YX.
Answer: YX is a glide reflection. The invariant line in this case is the line g through A perpendicular to b, so you would construct this line g. The vector for the translation is vector AA', where A' is the reflection of A in b, so you will construct A' also.
Reason: Let a be the line through A perpendicular to g. Then X = RaRg, so YX = (RbRa)Rg. But RbRa is the translation that takes A to A'.
Notice that vector AA' is parallel to g as required in the definition of glide reflection.Also notice that the translation RaRb is the inverse of translation RbRa. Also notice that X is both = RaRg and RgRa (check the angles in both cases).. Finally notice that in 3A and 3B that XY and YX are inverses of each other.
Example 4. Given a figure with points A and B. If X is rotation A90 and Y is rotation B90, what isometry is XY? Construct the defining data of XY.
Answer: XY is J180, where C is constructed as in problem 10-1.
Note: Since X or Y can be a line reflection, rotation, point reflection (special case of rotation), translation, or glide reflection, there are 5 possible choices for X and 5 for Y. You should be able to answer confidently for each case.