Question

Construct a generic, non-regular quadrilateral ABCD in GeoGebra. Then explore what happens when it is translated by a vector and then reflected by a line parallel to the vector. Notice that this composition is not name (Glide Reflection) and added to the list of "basic" isometries. However, since this type of isometry is defined via a composition, it is reasonable to ask if the order in which the two components are applied matters. #4 Is the composition of a translation followed by a reflection (over a line parallel to the vector) the same as first reflecting over the line and then applying the translation?

Answer 1

In the figure above, we have quadrilateral CDFE. It is first translated by a vector u leading to quadrilateral C'D'E'F' which is then reflection across line AB to get the quadrilateral at top right.

In second operation, CDFE is first reflected across the line yielding C1'D1'E1'F1' which is then translated by vector u. Again we get the quadrilateral at the top right.

Hence we see that in glide-reflection, the order of reflection and translation is immaterial. We get the same transformation irrespective of whether we translate first by a vector and then reflect across a line parallel to the vector or reflection across the line is carried out first and then translation by the vector is performed.