Chapter 4.3, Problem 14E

To determine

which of the lines are parallel or perpendicular.

Answer to Problem 14E

The lines $aandb$ are parallel to each other.

The lines a, b and c are not perpendicular

Explanation of Solution

Given:

  

Calculation:

Consider the lines $a,b,andc$ such that the coordinates forline $aare\left(-1,1\right)and\left(2,0\right)$ , the coordinates for line $bare\left(0,5\right)and\left(3,4\right)$ and the coordinates for line $care\left(0,0\right)and\left(2,5\right)$ .

The objective is to determine which lines are parallel or perpendicular.

To determine whether the lines are parallel or perpendicular, find the slopes of lines $a,b,andc$ .

The slope of a line is the ratio of difference of $ycoordinatetoxcoordinate$ .

Therefore, the slope of line $a$ is,

  $\begin{array}{l}{m}_a=\frac{1-0}{-1-2}\\ =\frac{1}{-3}\\ =-\frac{1}{3}\end{array}$

The slope of line $b$ is,

  $\begin{array}{l}{m}_b=\frac{5-4}{0-3}\\ =\frac{1}{-3}\\ =-\frac{1}{3}\end{array}$

The slope of line $c$ is,

  $\begin{array}{l}{m}_c=\frac{0-5}{0-2}\\ =\frac{-5}{-2}\\ =\frac{5}{2}\end{array}$

Since the slopes of all the three lines $aandb$ are equal, therefore, lines $aandb$ are parallel to each other.

Also, none of them are the negative inverse of each other, therefore they are not perpendicular

Conclusion:

The lines $aandb$ are parallel to each other.

The lines a, b and c are not perpendicular