Chapter 11.4, Problem 31E

(a)

To determine

To Prove: $KK\text{'}$ is perpendicular to $landm$ .

Explanation of Solution

Given:

  

Given: $l\parallel m$

Suppose if the reflection is maps $JKtoJ\text{'}K\text{'}$

Then

  $KK\text{'}$ is perpendicular to $l$ .

Now

If a reflection in $m$ maps to $J\text{'}K\text{'}toJ\text{'}\text{'}K\text{'}\text{'}$

Then $K\text{'}K\text{'}\text{'}$ is perpendicular to $m$ .

Since

  $l\parallel m$

Then $K\text{'}K\text{'}\text{'}$ is perpendicular to $landm$ .

Hence proved.

(b)

To determine

To Prove: $KK\text{'}=2d$

Explanation of Solution

Given:

  

Given: $l\parallel m$

  $\begin{array}{l}K{K}_l=K_{l}K\text{'}\\ K\text{'}{K}_m=K_{m}K\text{'}\text{'}\\ KK\text{'}\text{'}=K{K}_l+K_{l}K\text{'}+K\text{'}{K}_m+K_{m}K\text{'}\text{'}\\ KK\text{'}\text{'}=2K_{l}K\text{'}+2K_{m}K\text{'}\text{'}\\ KK\text{'}\text{'}=2\left(K_{l}K\text{'}+K_{m}K\text{'}\text{'}\right)\\ KK\text{'}\text{'}=2d\end{array}$

Hence proved.