Chapter 14.1, Problem 5WE

(a)

To determine

To plot: The three points $\text{A}\left(0,4\right),\text{B}\left(4,6\right)$ and $\text{C}\left(2,0\right)$ and their images.

Explanation of Solution

Given information:

The function is $T\left(x,y\right)=\left(x+4,y-2\right)$ .

Calculation:

Consider the function $T\left(x,y\right)=\left(x+4,y-2\right)$ .

To calculate the image of $\text{A}\left(0,4\right)$ , substitute 0 for $x$ and 4 for $y$ in the function.

  $\begin{array}{c}T\left(0,4\right)=\left(0+4,4-2\right)\\ =\left(4,2\right)\end{array}$

Similarly,

To calculate the image of $\text{B}\left(4,6\right)$ , substitute 4 for $x$ and 6 for $y$ in the function.

  $\begin{array}{c}T\left(4,6\right)=\left(4+4,6-2\right)\\ =\left(8,4\right)\end{array}$

To calculate the image of $\text{C}\left(2,0\right)$ , substitute 2 for $x$ and 0 for $y$ in the function.

  $\begin{array}{c}T\left(2,0\right)=\left(2+4,0-2\right)\\ =\left(6,-2\right)\end{array}$

So, the image of $\text{A}\left(0,4\right),\text{B}\left(4,6\right)$ and $\text{C}\left(2,0\right)$ are $A^{\text{'}}\left(4,2\right)$ , $B^{\text{'}}\left(8,4\right)$ and $C^{\text{'}}\left(6,-2\right)$ .

Plot points $\text{A}\left(0,4\right),\text{B}\left(4,6\right)$ and $\text{C}\left(2,0\right)$ .

  

(b)

To determine

To check: Whether the transformation is isometry or not.

Answer to Problem 5WE

Yes, the transformation is isometry.

Explanation of Solution

Calculation:

In part (a), the images and preimages are congruent. So, the transformation is isometry.

Therefore, the transformation is isometry.

(c)

To determine

To find: The pre image of 12 and 6 for the function $T\left(x,y\right)=\left(x+4,y-2\right)$ .

Answer to Problem 5WE

The pre image of 12 and 6 is $\left(8,8\right)$ .

Explanation of Solution

Calculation:

To calculate the pre image, equate the corresponding values.

  $\begin{array}{c}x+4=12\\ x=12-4\\ x=8\end{array}$

and,

  $\begin{array}{c}y-2=6\\ y=6+2\\ y=8\end{array}$

Therefore, the pre image of 12 and 6 is $\left(8,8\right)$ .