Chapter 14.1, Problem 22WE

(a)

To determine

To find: Whether the mapping $M$ appear to be an isometry or not.

Answer to Problem 22WE

Yes, the mapping $M$ appears to be an isometry.

Explanation of Solution

Given information: The given figure is as follows.

  

From the given figure, it can be seen that the mapping $M$ appear as an isometry because every point of the first region has its image in the second region and all the sets of pre-images and images are equidistance from the line k .

Therefore, the mapping $M$ appears to be an isometry.

(b)

To determine

To find: The image of point $P$ if it lies on the line $k$ .

Answer to Problem 22WE

The image of $P$ that lies on the line k is $P$ .

Explanation of Solution

Given information: The given figure is as follows.

  

The image of line k is also line k . If a point $P$ lies on the line $k$ then its image will also be on the line $k$ , because due to isometry the relative distance will be the same.

So, point $P$ will have a zero distance after mapping as well.

Therefore, the image of $P$ that lies on the line k is $P$ .

(c)

To determine

To find: The relation between line $k$ and $\overline{Q{Q}^{\prime }}$ .

Answer to Problem 22WE

The line $k$ and $\overline{Q{Q}^{\prime }}$ are perpendicular to each other.

Explanation of Solution

Given information: The given figure is as follows.

  

It is given that point $Q$ is not on line $k$ , and $M\left(Q\right)=Q^{\prime }$ .

Consider another point $Q$ on the left side of the line k .

It is given that $M\left(Q\right)=Q^{\prime }$ . This means that the image of point $Q$ on the other side of the line is $Q^{\prime }$ . Both the points are at the same distance from line k . So, it form a straight line, then the line $\overline{QQ\text{'}}$ will be perpendicular to the line $k$ .

Therefore, the line $k$ and $\overline{QQ\text{'}}$ are perpendicular to each other.