Chapter 14.3, Problem 13WE

To determine

To find:Translation that map $P$ to $P\text{'}\text{'}$ .

Answer to Problem 13WE

The translation that maps $P\text{'}$ to $P\text{'}\text{'}$

  $T:(x,y)\to (x-4,y+9)$ .

Explanation of Solution

Given information :

$R$ map point $P$ to $P\text{'}$ and $S$ map point $P\text{'}$ to $P\text{'}\text{'}$ .

  $\begin{array}{l}R:(x,y)\to (x+1,y+2)\\ S:(x,y)\to (x-5,y+7)\end{array}$

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.

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.

As per the given problem

$R$ map point $P$ to $P\text{'}$ and $S$ map point $P\text{'}$ to $P\text{'}\text{'}$ .

  $\begin{array}{l}R:(x,y)\to (x+1,y+2)\\ S:(x,y)\to (x-5,y+7)\end{array}$

$T$ is composition of $R$ and $S$ .

  $\begin{array}{l}T(x,y)=S(R(x,y))\\ \Rightarrow S(x+1,y+2)=((x+1)-5,(y+2)-7)\\ \Rightarrow (x-4,y+9)\\ \therefore T:(x,y)\to (x-4,y+9)\end{array}$

Hence,

The translation that maps $P\text{'}$ to $P\text{'}\text{'}$

  $T:(x,y)\to (x-4,y+9)$ .