Chapter 13, Problem 29RP

To determine

To find: The length of the altitude from C to AB line.

Answer to Problem 29RP

Thelength of the altitude is $2\sqrt{10}$ .

Explanation of Solution

Given:

The given points are $A\left(0,-17\right),B\left(4,-4\right),C\left(12,-1\right)$ .

Calculation:

From the given points the slope of the line AB is calculated as,

  $\begin{array}{c}m=\frac{-5-\left(-17\right)}{4-0}\\ =\frac{12}{4}\\ =3\end{array}$

Consider the slope point equation is of the form,

  $y=mx+b$

Then the equation of the line AB is,

  $y=3x-17$

Calculate the slope of the altitude as,

  $y=\frac{-1}{3}$

Then, the equation of the altitude is,

  $\begin{array}{l}y=-\frac{1}{3}x+c\\ -1=-\frac{1}{3}\left(12\right)+c\\ c=3\end{array}$

Then, the equation is,

  $y=-\frac{1}{3}x+3$

Then, the intersection of the two lines occurs at,

  $\begin{array}{l}3x-17=-\frac{1}{3}x+3\\ x=6\end{array}$

Then,

  $\begin{array}{c}y=-\frac{1}{3}\left(6\right)+3\\ =1\end{array}$

Thus, the intersection of the two points occurs at $\left(6,1\right)$ .

The length of the altitude is obtained as,

  $\begin{array}{c}\sqrt{{\left(12-6\right)}^2+{\left(-1-1\right)}^2}=\sqrt{36+4}\\ =\sqrt{40}\\ =2\sqrt{10}\end{array}$