Chapter 14.3, Problem 22WE

To determine

To proof:Image of point glide followed by reflection will be different from the same point image where reflection followed by glide when glide is not parallel to line of reflection.

Answer to Problem 22WE

Image of point glide followed by reflection will be different from the same point image where reflection followed by glide when glide is not parallel to line of reflection.

Explanation of Solution

Given information :

Glide is not parallel to line of reflection.

Concept used:

A translation moves ("slides") an object a fixed distance in a given direction without changing its size or shape, and without turning it or flipping it.The original object is called the pre-image, and the translation is called the image.

Glide is not parallel to line of reflection.

For the proof we will solve by example.

Let the point be $P(x,y)$ .

Translation, $T:(x,y)\to (x+2,y)$

Line of reflection, $R_m:y=x$

Case 1

Image of point when glide followed by reflection.

After glide,

  $T:(P)=(x,y)\to (x+2,y)=P\text{'}$

After reflection, where $(x,y)$ map to $(y,x)$

  $R_m:(P")=(x+2,y)\to (y,x+2)=P\text{'}\text{'}$

Image of point $P(x,y)$ is $P"=(y,x+2)$ ……………. (I)

Case 2

Image of point when reflection followed by glide.

After reflection, where $(x,y)$ map to $(y,x)$

  $R_m:(P)=(x,y)\to (y,x)=P\text{'}$

After glide,

  $T:(P\text{'})=(y,x)\to (y+2,x)=P\text{'}\text{'}$

Image of point $P(x,y)$ is $P"=(y+2,x)$ …………………. (II)

From I & II, it is shown that image of point reflection followed by glide is not same compared to glide followed by reflection.

Hence,

Image of point glide followed by reflection will be different from the same point image where reflection followed by glide when glide is not parallel to line of reflection.