Chapter 14.1, Problem 6WE

(a).

To determine

To graph: The three points $A\left(0,4\right)$ , $B\left(4,6\right)$ and $C\left(2,0\right)$ and their images under the given transformation.

Explanation of Solution

Given information: The given transformation is $S:\left(x,y\right)\to \left(2x+4,2y-2\right)$ .

Graph:

To find the images of the given points, consider the given transformation:

  $S:\left(x,y\right)\to \left(2x+4,2y-2\right)$

To determine image of point $A\left(0,4\right)$ , substitute $0$ for $x$ and $4$ for $y$ in the above expression.

  $\left(2\left(0\right)+4,2\left(4\right)-2\right)=\left(4,6\right)$

So, image of point $A\left(0,4\right)$ is $A^{\prime }\left(4,6\right)$ .

To determine image of point $B\left(4,6\right)$ , substitute $4$ for $x$ and $6$ for $y$ in the above expression.

  $\left(2\left(4\right)+4,2\left(6\right)-2\right)=\left(12,10\right)$

Therefore, image of point $B\left(4,6\right)$ is $B^{\prime }\left(12,10\right)$ .

To determine image of point $C\left(2,0\right)$ , substitute $2$ for $x$ and $0$ for $y$ in the above expression.

  $\left(2\left(2\right)+4,2\left(0\right)-2\right)=\left(8,-2\right)$

So, the image of point $C\left(2,0\right)$ is $C^{\prime }\left(8,-2\right)$ .

Plot and connect the ordered pair $A\left(0,4\right)$ , $B\left(4,6\right)$ and $C\left(2,0\right)$ and their images $A^{\prime }\left(4,6\right)$ , $B^{\prime }\left(12,10\right)$ and $C^{\prime }\left(8,-2\right)$ on the coordinate graph as shown below:

Interpretation:

In the above graph, $A^{\prime }\left(4,6\right)$ is the image of point $A\left(0,4\right)$ , $B^{\prime }\left(12,10\right)$ is the image of point $B\left(4,6\right)$ and $C^{\prime }\left(8,-2\right)$ is the image of point $C\left(2,0\right)$ .

(b).

To determine

To find: Whether the given transformation appears to be an isometric or not.

Answer to Problem 6WE

The given transformation appears not to be an isometric.

Explanation of Solution

Given information: The given transformation is $S:\left(x,y\right)\to \left(2x+4,2y-2\right)$ .

Calculation:

As calculated in the above graph, $A^{\prime }\left(4,6\right)$ is the image of point $A\left(0,4\right)$ , $B^{\prime }\left(12,10\right)$ is the image of point $B\left(4,6\right)$ and $C^{\prime }\left(8,-2\right)$ is the image of point $C\left(2,0\right)$ .

In the theorem of transformations, an isometry maps a triangle to a congruent triangle.

For mapping the distances between image and point must be same such as $\overline{AB}\cong \overline{A^{\prime }{B}^{\prime }}$

, $\overline{BC}\cong \overline{B^{\prime }{C}^{\prime }}$ and $\overline{AC}\cong \overline{A^{\prime }{C}^{\prime }}$ .If they have the equal distance, then this implies that two triangle are congruent and thus isometric.

The expression for distance formula is:

  $\text{distance}=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$

Let, the point of $A\left(0,4\right)=\left(x_1,y_1\right)$ and $B\left(4,6\right)=\left(x_2,y_2\right)$ .

To determine the distance $AB$ , substitute $0$ for $x_1$ , $4$ for $y_1$ , $4$ for $x_2$ and $6$ for $y_2$ in the expression for distance formula.

  $\begin{array}{c}AB=\sqrt{{\left(4-0\right)}^2+{\left(6-4\right)}^2}\\ =\sqrt{{\left(4\right)}^2+{\left(2\right)}^2}\\ =\sqrt{16+4}\\ =\sqrt{20}\end{array}$

Let, the point of $A^{\prime }\left(4,6\right)=\left(x_1,y_1\right)$ and $B^{\prime }\left(12,10\right)=\left(x_2,y_2\right)$ .

To determine the distance $A^{\prime }{B}^{\prime }$ , substitute $4$ for $x_1$ , $6$ for $y_1$ , $12$ for $x_2$ and $10$ for $y_2$ in the expression for distance formula.

  $\begin{array}{c}{A}^{\prime }{B}^{\prime }=\sqrt{{\left(12-4\right)}^2+{\left(10-6\right)}^2}\\ =\sqrt{{\left(8\right)}^2+{\left(4\right)}^2}\\ =\sqrt{64+16}\\ =\sqrt{80}\end{array}$

Let, the point of $B\left(4,6\right)=\left(x_1,y_1\right)$ and $C\left(2,0\right)=\left(x_2,y_2\right)$ .

To determine the distance $BC$ , substitute $4$ for $x_1$ , $6$ for $y_1$ , $2$ for $x_2$ and $0$ for $y_2$ in the expression for distance formula.

  $\begin{array}{c}BC=\sqrt{{\left(2-4\right)}^2+{\left(0-6\right)}^2}\\ =\sqrt{{\left(-2\right)}^2+{\left(-6\right)}^2}\\ =\sqrt{4+36}\\ =\sqrt{40}\end{array}$

Let, the point of $B^{\prime }\left(12,10\right)=\left(x_1,y_1\right)$ and $C^{\prime }\left(8,-2\right)=\left(x_2,y_2\right)$ .

To determine the distance $B^{\prime }{C}^{\prime }$ , substitute $12$ for $x_1$ , $10$ for $y_1$ , $8$ for $x_2$ and $-2$ for $y_2$ in the expression for distance formula.

  $\begin{array}{c}{B}^{\prime }{C}^{\prime }=\sqrt{{\left(8-12\right)}^2+{\left(-2-10\right)}^2}\\ =\sqrt{{\left(-4\right)}^2+{\left(-12\right)}^2}\\ =\sqrt{16+144}\\ =\sqrt{160}\end{array}$

Let, the point of $A\left(0,4\right)=\left(x_1,y_1\right)$ and $C\left(2,0\right)=\left(x_2,y_2\right)$ .

To determine the distance $AC$ , substitute $0$ for $x_1$ , $4$ for $y_1$ , $2$ for $x_2$ and $0$ for $y_2$ in the expression for distance formula.

  $\begin{array}{c}AC=\sqrt{{\left(2-0\right)}^2+{\left(0-4\right)}^2}\\ =\sqrt{{\left(2\right)}^2+{\left(-4\right)}^2}\\ =\sqrt{4+16}\\ =\sqrt{20}\end{array}$

Let, the point of $A^{\prime }\left(4,6\right)=\left(x_1,y_1\right)$ and $C^{\prime }\left(8,-2\right)=\left(x_2,y_2\right)$ .

To determine the distance $A^{\prime }{C}^{\prime }$ , substitute $4$ for $x_1$ , $6$ for $y_1$ , $8$ for $x_2$ and $-2$ for $y_2$ in the expression for distance formula.

  $\begin{array}{c}{A}^{\prime }{C}^{\prime }=\sqrt{{\left(8-4\right)}^2+{\left(-2-6\right)}^2}\\ =\sqrt{{\left(4\right)}^2+{\left(-8\right)}^2}\\ =\sqrt{16+64}\\ =\sqrt{80}\end{array}$

Therefore, from the above it is prove that $\overline{AB}\ne \overline{A^{\prime }{B}^{\prime }}$ , $\overline{BC}\ne \overline{B^{\prime }{C}^{\prime }}$ and $\overline{AC}\ne \overline{A^{\prime }{C}^{\prime }}$ . Since, they don’t have the equal distance, this implies that two triangle are not congruent and thus not isometric.

(c).

To determine

To find: The coordinates of the preimage of the given point.

Answer to Problem 6WE

The coordinates of the preimage of the given point is $\left(4,4\right)$ .

Explanation of Solution

Given information: The given transformation is $S:\left(x,y\right)\to \left(2x+4,2y-2\right)$ and the point of which preimage is to be determined is $\left(12,6\right)$ .

Calculation:

The given transformation is $S:\left(x,y\right)\to \left(2x+4,2y-2\right)$ and $S\left(x,y\right)\to \left(12,6\right)$

On comparing the two values of $x-\text{axis}$ :

  $\begin{array}{c}2x+4=12\\ 2x=12-4\\ 2x=8\\ x=4\end{array}$

Subtract $4$ from both sides of the above expression.

  $\begin{array}{c}2x+4-4=12-4\\ 2x=8\end{array}$

To determine the value of $x$ , divide both the sides of the above expression by $2$ .

  $\begin{array}{c}2x÷2=8÷2\\ x=4\end{array}$

On comparing the two values of $y-\text{axis}$ :

  $\begin{array}{c}2y-2=6\\ 2y=6+2\\ 2y=8\\ y=4\end{array}$

Therefore, the coordinates of the preimage of $\left(12,6\right)$ is $\left(4,4\right)$ .