Chapter 14.5, Problem 25WE

a.

To determine

To show that the slopes of $\overline{PQ}\text{ and }\overline{P\text{'}Q\text{'}}$ are equal.

Explanation of Solution

Given information:

The dilation $D_{O,k}$ maps $\overline{PQ}\text{ to }\overline{P\text{'}Q\text{'}}$ .

  

Formula used:

The following formula is used:

  $m=\frac{y_2-y_1}{x_2-x_1}$

Proof:

Calculate the slope of $\overline{PQ}$ :

  $m=\frac{y_2-y_1}{x_2-x_1}$

Calculate the slope of $\overline{P\text{'}Q\text{'}}$ ;

  $\begin{array}{l}m=\frac{k{y}_2-k{y}_1}{k{x}_2-k{x}_1}\\ m=\frac{k\left(y_2-y_1\right)}{k\left(x_2-x_1\right)}\\ m=\frac{y_2-y_1}{x_2-x_1}\end{array}$

Hence, the slopes of $\overline{PQ}\text{ and }\overline{P\text{'}Q\text{'}}$ are equal.

b.

To determine

To complete the statement.

Answer to Problem 25WE

Part (a) proves that $\overline{PQ}\text{ and }\overline{P\text{'}Q\text{'}}$ are parallel.

Explanation of Solution

Given information:

The dilation $D_{O,k}$ maps $\overline{PQ}\text{ to }\overline{P\text{'}Q\text{'}}$ .

  

The given statement is:

“Part (a) proves that $\overline{PQ}\text{ and }\overline{P\text{'}Q\text{'}}$ are_____”

Since the slope of both the lines are same.

Part (a) proves that $\overline{PQ}\text{ and }\overline{P\text{'}Q\text{'}}$ are parallel.