Chapter 13.2, Problem 13PSB

To determine

To write: The point slope form and an equation of a line through $C$ parallel to $\overleftrightarrow{\text{AB}}$ .

Answer to Problem 13PSB

The point slope form of an equation of a line through $C$ parallel to $\overleftrightarrow{\text{AB}}$ is $y=\frac{1}{7}x+\frac{80}{7}$ .

Explanation of Solution

Given:

The given figure is as follows.

  

Calculation:

Point-slope formula is given by $m=\frac{y_2-y_1}{x_2-x_1}$ .

Substitute $2$ for $x_1$ , $16$ for $x_2$ , $3$ for $y_2$ and $1$ for $y_1$ in the equation $m=\frac{y_2-y_1}{x_2-x_1}$ .

  $\begin{array}{c}m=\frac{3-1}{16-2}\\ =\frac{2}{14}\\ =\frac{1}{7}\end{array}$

Determine the equation of line parallel to $\overleftrightarrow{\text{AB}}$ and passing through point $C\left(4,12\right)$ .

Substitute $4$ for $x_1$ , $12$ for $y_1$ , $\frac{1}{7}$ for $m$ in the equation $y-y_1=m\left(x-x_1\right)$ .

  $\begin{array}{c}y-12=\frac{1}{7}\left(x-4\right)\\ 7y-84=x-4\\ y=\frac{1}{7}x+\frac{80}{7}\end{array}$

Therefore, the point slope form of an equation of a line through $C$ parallel to $\overleftrightarrow{\text{AB}}$ is $y=\frac{1}{7}x+\frac{80}{7}$ .