Chapter 9.4, Problem 30PPS

To determine

To Prove: To prove one case of composition of isomerise theorem.

Explanation of Solution

Given:

The translational along $\left(a,b\right)$ maps $X$ to $X^{\prime }$ and $Y$ to $Y^{\prime }$ then prove that $\overline{XY}\cong \overline{X^{\prime }{Y}^{\prime }}$ .

The given figure is shown in Figure 1

  

Figure 1

Calculation:

Consider that the translation moves the figure with all the points in the same distance and direction, therefore the points X and Y move the same distance in the same direction and $\overline{XY}\cong \overline{X^{\prime }{Y}^{\prime }}$ .

Consider that the reflection in a line in which each of the point of the pre-image and its corresponding point on the image are same distance from the line, thus the points $X^{″}$ and $Y^{″}$ are at the same distance as the line z as $X^{\prime }$ and $Y^{\prime }$ are. Therefore, $\overline{X^{\prime }{Y}^{\prime }}\cong \overline{X^{″}{Y}^{″}}$ .

Hence, proved.