Chapter 13.2, Problem 15PSB

To determine

To write: The equation of the altitude from $\text{C}$ to $\overleftrightarrow{\text{AB}}$ .

Answer to Problem 15PSB

The equation of the altitude from $\text{C}$ to $\overleftrightarrow{\text{AB}}$ is $y=-7x+40$ .

Explanation of Solution

Given:

The given figure is as follows.

  

Calculation:

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

Find the slope of line $\overleftrightarrow{\text{AB}}$ .

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

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

Take reciprocal of $\frac{1}{7}$ since, it is required to find the altitude from $\text{C}$ to $\overleftrightarrow{\text{AB}}$ . So

  $m=-7$

Determine the equation of the altitude from $\text{C}$ to $\overleftrightarrow{\text{AB}}$ .

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

  $\begin{array}{c}y-12=-7\left(x-4\right)\\ y-12=-7x+28\\ y=-7x+40\end{array}$

Therefore, the equation of the altitude from $\text{C}$ to $\overleftrightarrow{\text{AB}}$ is $y=-7x+40$ .