Chapter 8.1, Problem 5GP

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 $\left(-1,-1\right)$

Equation of perpendicular bisector is $9y=-4x-13$ .

Given: two points $\left(3,8\right),\left(-5,-10\right)$

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 $\left(3,8\right),\left(-5,-10\right)$ :

  $\begin{array}{c}\left(\frac{3+(-5)}{2},\frac{8+(-10)}{2}\right)=\left(\frac{-2}{2},\frac{-2}{2}\right)\\ =\left(-1,-1\right)\end{array}$

Now slope of the points $\left(3,8\right),\left(-5,-10\right)$ :

  $\begin{array}{c}\frac{-10-8}{-5-3}=-\frac{-18}{-8}\\ =\frac{9}{4}\end{array}$

Slope of perpendicular lines are negative reciprocal of each other.

Slope of perpendicular line: $\frac{-1}{\frac{9}{4}}=-\frac{4}{9}$

Equation of a line:

  $\begin{array}{c}y-(-1)=\frac{-4}{9}(x-(-1))\\ y+1=\frac{-4}{9}x-\frac{4}{9}\\ y=\frac{-4}{9}x-\frac{4}{9}-1\\ y=\frac{-4x-4-9}{9}\\ 9y=-4x-13\end{array}$

Conclusion:

Midpoint is $\left(-1,-1\right)$

Equation of perpendicular bisector is $9y=-4x-13$ .