Chapter 9.4, Problem 38HP

To determine

To find: Whether any points are invariant under the glide reflections, under compositions of transformations.

Answer to Problem 38HP

The transformation is not a glide reflection.

Explanation of Solution

Consider that Daniel and Lolita are translating triangle XYZ along the coordinates $\langle 2,2\rangle$ and reflects it in the line $y=2$ . Daniel says that the transformation is glide reflection. Then, the Lolita disagrees and says that the transformation is the composition of the transformation.

The first is to translate triangle XYZ along $\langle 2,2\rangle$ as,

  $\begin{array}{l}\left(x,y\right)=\left(x+2,y+2\right)\\ X\left(-3,-2\right)=X^{\prime }\left(-1,0\right)\\ Y\left(0,0\right)\to Y^{\prime }\left(2,2\right)\\ Z\left(1,-2\right)\to Z^{\prime }\left(3,0\right)\end{array}$

Consider the translational over the line $y=2$ as shown below,

  $\begin{array}{l}{X}^{\prime }\left(-1,0\right)\to X^{″}\left(2,2\right)\\ Z^{\prime }\left(3,0\right)\to Z^{″}\left(3,4\right)\end{array}$

Thus, the transformation is the composition of the translational and reflection, thus it is a composition of transformation. This shows that Lolita is correct and line $y=2$ is not parallel to the vector $\langle 2,2\rangle$ , thus the transformation is not a glide reflection.