Chapter 9.4, Problem 39HP

To determine

To find: Whether any points are invariant under the glide reflections and under the composition of transformations.

Answer to Problem 39HP

The invariant points in the graph and the points may remain invariant under the composition of the transformation. There may be an invariant point in the figure as it is rotated or reflected, rotated twice and reflected twice.

Explanation of Solution

Consider the glide reflection shown below in Figure 1

  

Figure 1

There are no invariant points in the graph and the points may remain invariant under the composition of the transformation. There may be an invariant point in the figure as it is rotated or reflected, rotated twice and reflected twice.

Consider the diagram for the two 180 degree rotation as shown in Figure 2

  

Figure 2

Consider the figure for two reflection about the same line as shown in Figure 3

  

Figure 3

The image can be obtained back to the original image by rotation and reflection. Thus, the triangle ABC is rotated 240 degree about the origin and then reflected twice about the x axis and in the dashed line. Thus, in this way the final image is mapped to the original image as shown in Figure 4

  

Figure 4