Chapter 9.6, Problem 32PPS

(a)

To determine

To Graph: The dilation of $DEF$ centered at the point $D$ with the scale factor $3$ .

Explanation of Solution

Given:

  

Here,

  $\begin{array}{l}D(1,2)\\ E(3,2)\\ F(3,0)\\ Fordilation\\ DE\text{'}=3DE\\ DF\text{'}=3DF.\end{array}$

The image of graph after dilation is shown below.

  

(b)

To determine

To Describe: The dilation as a composition of transformations including dilation with the scale factor of three centered at the origin.

Answer to Problem 32PPS

The composition of dilation with the scale factor three centered at the origin and the translation along $(-2,-4)$ .

Explanation of Solution

Given:

  

Here,

  $\begin{array}{l}D(1,2)=D\text{'}(3,6)\\ E(3,2)=E\text{'}(9,6)\\ F(3,0)=F\text{'}(9,0)\end{array}$

The image of graph is shown below.

  

(c)

To determine

To Describe: If the figure is dilated by the scale factor three with the center of dilation $(x,y)$ . What composition including the dilation with the scale factor $3$ centered at the origin, will produce the same.

Answer to Problem 32PPS

The composition of dilation with the scale factor three centered at the origin and the translation along $(-2x,-2y)$ .

Explanation of Solution

Given:

  

The center of dilation is at $D(1,2)$ .

From the above

  $\begin{array}{l}when\\ x=1,y=2\end{array}$

The translation vector is $<-2,-4>$

Now finding the translation vector in terms of $xandy$ .

Thus

The translation vector yields $<-2x,-2y>whenx=1,y=2.$

Thus

  $(-2x,-2y)$