Chapter 8.1, Problem 3GP

Solution

To find: for the line segment joining the two given points, (a) find the midpoint and (b) write an equation for the perpendicular bisector.

Midpoint is $\left(-2,6\right)$

Equation of perpendicular bisector is $3y=x+20$ .

Given: two points $\left(0,0\right),\left(-4,12\right)$

Concept Used:

Midpoint formula is given by $\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$.

Slope of a line $(m)=\frac{y_2-y_1}{x_2-x_1}$

Equation of point slope-form: $y-y_1=m\left(x-x_1\right)$

And Slope of perpendicular lines are negative reciprocal of each other.

Explanation:

Finding the midpoint of $\left(0,0\right),\left(-4,12\right)$ :

  $\begin{array}{c}\left(\frac{0+(-4)}{2},\frac{0+12}{2}\right)=\left(\frac{-4}{2},\frac{12}{2}\right)\\ =\left(-2,6\right)\end{array}$

Now slope of the points $\left(0,0\right),\left(-4,12\right)$ :

  $\begin{array}{c}\frac{12-0}{-4-0}=-\frac{12}{4}\\ =-3\end{array}$

Slope of perpendicular lines are negative reciprocal of each other.

Slope of perpendicular line: $\frac{-1}{-3}=\frac{1}{3}$

Equation of a line:

  $\begin{array}{c}y-6=\frac{1}{3}(x+2)\\ y-6=\frac{1}{3}x+\frac{2}{3}\\ y=\frac{1}{3}x+\frac{2}{3}+6\\ y=\frac{x+2+18}{3}\\ 3y=x+20\end{array}$

Conclusion:

Midpoint is $\left(-2,6\right)$

Equation of perpendicular bisector is $3y=x+20$ .