Chapter 14.6, Problem 11CE

To determine

To prove that the composite of two isometries is an isometry.

Answer to Problem 11CE

  $\text{So }T:\overline{PQ}\to \overline{PQ}.$

Explanation of Solution

Given information: Given transformation $S$ to another transformation $T$ can be written as $SoT.$

Calculation:

Let $S$ and $T$ be isometrics, now consider a line $PQ.$

Under the transformation $S$ , the image of the line $PQ$ is $S:\overline{PQ}\to \overline{P^{\prime }{Q}^{\prime }}.$

And because $S$ is an isometric, therefore; $S:\overline{PQ}\to \stackrel{-}{PQ}\mathrm{.....}\left(1\right)$

Similarly, Under the transformation $S$ , the image of the line $PQ$ is : $T:\overline{PQ}\to \overline{P^{\prime }{Q}^{\prime }}.$

And because $S$ is an isometric, therefore; $T:\overline{PQ}\to \overline{PQ}\mathrm{.....}(2)$

Use $(1)$ and $(2)$ and find out the composite of the transformations;

  $\text{So }T:\overline{PQ}\to \overline{PQ}.$

Hence, the composite of two isometries is an isometry.