Chapter 14.6, Problem 29WE

To determine

To write: The type of transformation of the composite dilations $D_{B,\frac{1}{2}}\circ D_{A,2}$ .

Answer to Problem 29WE

The composite of both dilations is a translation.

Explanation of Solution

Given information: The $D_{A,2}:\overline{PQ}\to \overline{P^{\prime }{Q}^{\prime }}$ , and $D_{B,\frac{1}{2}}:\overline{P^{\prime }{Q}^{\prime }}\to \overline{P^{″}{Q}^{″}}$ .

  

Calculation:

A transformation related to similarity rather than congruence is called dilation. The dilation $D_{O,k}$ has center $O$ and nonzero scale factor $k$ .

The dilation $D_{O,k}$ maps any point $P$ to a point $P^{\prime }$ determined as follows.

(1) If $k>1$ , $P^{\prime }$ lie on $\overline{OP}$ and $O{P}^{\prime }=k\cdot OP$ .

(2) If $k<0$ , $P^{\prime }$ lies on the ray opposite $\overline{OP}$ and $O{P}^{\prime }=\left|k\right|\cdot OP$ .

(3) The center $O$ is its own image.

The dilation is called an expansion for $\left|k\right|>1$ and the dilation is called an contraction for $\left|k\right|<1$ .

A glide or translation is the movement of a plane from one position to another without changing its orientation, or in simple words, a transformation that glides all points of a plane the same distance in the same direction is called a translation.

Consider dilation about the point $A$ and with scalar factor of $2$ such as;

  $D_{A,2}:\overline{PQ}\to \overline{P^{\prime }{Q}^{\prime }}$

Consider dilation about the point $B$ and with scalar factor of $\frac{1}{2}$ such as;

  $D_{B,\frac{1}{2}}:\overline{P^{\prime }{Q}^{\prime }}\to \overline{P^{″}{Q}^{″}}$

The first dilation is an example of expansion and it expands the line $PQ$ to $P^{\prime }{Q}^{\prime }$ with scalar factor of $2$ , whereas the second dilation is an example of contraction and it contracts the line $P^{\prime }{Q}^{\prime }$ to $P^{″}{Q}^{″}$ with scalar factor of $\frac{1}{2}$ .

It is clear that it composite of both dilations can be written as;

  $D_{B,\frac{1}{2}}\circ D_{A,2}:\overline{PQ}\to \overline{P^{″}{Q}^{″}}$

The only difference is the position of the image, and it is not same, so it is a case of translation.

Therefore, the composite of both dilations is a translation.