Chapter 4.3, Problem 10Q

To determine

To calculate: To determine which of the lines are parallel or perpendicular

Answer to Problem 10Q

Line $a$ and line $b$ are parallel.

Line $b$ and line $c$ are perpendicular

Line $a$ and line $c$ are perpendicular

Explanation of Solution

Given: Equation of three lines as below:

Line a passes through $\left(-2,2\right)$ and $\left(2,1\right)$

Line b passes through $\left(1,-8\right)$ and $\left(3,0\right)$

Line a passes through $\left(-4,-3\right)$ and $\left(0,-2\right)$

Formula Used:

Slope of line passing through two points given as

  $y=\frac{y_2-y_1}{x_2-x_1}$

1. When two lines are parallel, then the slopes of lines are equal
2. When two lines are perpendicular, then the product of slopes of line is equal to $-1$

Calculation:

Slope of line passing through points $\left(-2,2\right)$ and $\left(2,1\right)$ is calculated as

  $\begin{array}{l}y=\frac{-2-2}{2-1}\\ y=\frac{-4}{1}\\ y=-4\end{array}$

Slope of line passing through points $\left(1,-8\right)$ and $\left(3,0\right)$ is calculated as

  $\begin{array}{l}y=\frac{-8-0}{3-1}\\ y=\frac{-8}{2}\\ y=-4\end{array}$

Slope of line passing through points $\left(-4,-3\right)$ and $\left(0,-2\right)$ is calculated as

  $\begin{array}{l}y=\frac{-3-\left(-2\right)}{-4+0}\\ y=\frac{-3+2}{-4}\\ y=\frac{1}{4}\end{array}$

Since the slopes of line $a$ and line $b$ are equal

Thus, line $a$ and line $b$ are parallel.

Also, since product of slopes of line $a$ and line $c$ is $-1$

Thus, the line $a$ and line $c$ are perpendicular

Also, since product of slopes of line $b$ and line $c$ is $-1$

Thus, the line $b$ and line $c$ are perpendicular

Conclusion:

Hence, line $a$ and line $b$ are parallel.

Line $b$ and line $c$ are perpendicular

Line $a$ and line $c$ are perpendicular