Chapter 8.1, Problem 34E

Solution

The equation of the perpendicular bisector of the line segment joining the given two points.

  $y=\frac{1}{2}x-\frac{11}{4}$

Given information:

The points given are $\left(1,4\right)\text{ and }\left(6,-6\right)$ .

Formula used:

1. The slope m of a line passing through the points $\left(x_1,y{1}_{}\right),\text{ and }\left(x_2,y{2}_{}\right)$ , is given by: $m=\frac{y{2}_{}-y{1}_{}}{x_2-x_1}$ and slope of the perpendicular bisector:
2. The midpoint of a line passing through the points is given by: $\left(x,y\right)=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$
3. The equation of perpendicular bisector of the line joining two point and slope m is where c can be find by putting the middle point in this equation.

Calculation:

To calculate the equation of perpendicular bisector of the line joining two points $\left(1,4\right)\text{ and }\left(6,-6\right)$

First find the middle point of these two points using the formula

  

As,

  $\begin{array}{c}\left(x,y\right)=\left(\frac{1+6}{2},\frac{4+\left(-6\right)}{2}\right)\\ =\left(\frac{7}{2},\frac{4-6}{2}\right)\\ =\left(\frac{7}{2},\frac{-2}{2}\right)\\ =\left(\frac{7}{2},-1\right)\end{array}$

Now find the slope of the line passing through these points

  $\begin{array}{c}m=\frac{y{2}_{}-y{1}_{}}{x_2-x_1}\\ =\frac{-6-4}{6-1}\\ =\frac{-10}{5}\\ =-2\end{array}$

Thus ,the slope of the perpendicular bisector is

  $\begin{array}{c}-\frac{1}{m}=-\frac{1}{\left(-2\right)}\\ =\frac{1}{2}\end{array}$

Thus the equation of perpendicular bisector is

  $\begin{array}{c}y=\frac{-1}{m}x+c\\ =\frac{1}{2}x+c\end{array}$

Now find c by putting middle point in above equation

  $\begin{array}{c}y=-\frac{1}{4}x+c\\ -1=\frac{1}{2}\left(\frac{7}{2}\right)+c\\ -1=\frac{7}{4}+c\\ c=-1-\frac{7}{4}\\ c=\frac{-4-7}{4}\\ c=\frac{-11}{4}\end{array}$

Putting value of $c=\frac{-11}{4}$

  $\begin{array}{l}y=\frac{1}{2}x+\left(-\frac{11}{4}\right)\\ y=\frac{1}{2}x-\frac{11}{4}\end{array}$

Thus the required equation of perpendicular bisector is

  $y=\frac{1}{2}x-\frac{11}{4}$