Chapter 14.3, Problem 8CE

a.

To determine

To find:Image of $S(-1,3)$ under $T$

Answer to Problem 8CE

The image, point $S\text{'}=(1,5)$ of $S(-1,3)$ under $T$ .

Explanation of Solution

Given:

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

Concept Used:

In 2-dimensional geometry, a glide reflection is a symmetry operation that consists of a reflection over a line and then translation along that line, combined into a single operation.

Calculation:

As per the given problem

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

$x$ moves right 2 units & $y$ moves 2 unit up.

  $T(-1,3)\to (-1+2,3+2)=(1,5)$

Hence,

The image, point $S\text{'}=(1,5)$

b.

To determine

To find:Image point $S"$ of $S\text{'}$ under $R_m$

Answer to Problem 8CE

The image of $S\text{'}$ under $R_m$ , point $S"$ is $(5,1)$ .

Explanation of Solution

Given:

The line of reflection is line $m$ with equation $y=x$

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.

Calculation:

As per the given problem

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

  $T:S=(-1,3)\to (1,5)=S\text{'}$

Under the reflection in the line $y=x$ , the point $(x,y)$ is mapped to the point $(y,x)$ .

  $\begin{array}{l}{R}_m:(x,y)\to (y,x)\\ \therefore R_m:(1,5)\to (5,1)\\ \Rightarrow S"=(5,1)\end{array}$

Hence,

The image of $S\text{'}$ under $R_m$ , point $S"$ is $(5,1)$ .

c.

To determine

To find:Under glide reflection, $(x,y)$ is first mapped to $(?,?)$ and then to $(?,?)$ .

Answer to Problem 8CE

The glide reflection, $(x,y)$ is first mapped to $(1,5)$ and then to $(5,1)$ .

Explanation of Solution

Given:

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

The line of reflection is line $m$ with equation $y=x$

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.

Calculation:

As per the given problem

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

Under the reflection in the line $y=x$ , the point $(x,y)$ is mapped to the point $(y,x)$ .

  $R_m:(x,y)\to (y,x)$

  $S(-1,3)\to S\text{'}(1,5)\to S"(5,1)$

Hence,

The glide reflection, $(x,y)$ is first mapped to $(1,5)$ and then to $(5,1)$ .