Chapter B, Problem 45E

To determine

To calculate: To find the equation of the perpendicular bisector of the line segment joining the points $A\left(1,4\right)$ and $B\left(7,-2\right)$

Answer to Problem 45E

The equation of perpendicular bisector is $y=x-3$

Explanation of Solution

Given information: Points are $A\left(1,4\right)$ and $B\left(7,-2\right)$

Formula Used:

The mid-point of the line segment from $P_1\left(x_1,y_1\right)$ to $P_2\left(x_2,y_2\right)$ is $\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}$

Product of the slope of perpendicular line is equal to $-1$

Perpendicular bisector of a line segment is a line that is perpendicular to the line segment and passes through the mid-point of the line segment

Slope of the line passing through the points $P_1\left(x_1,y_1\right)$ to $P_2\left(x_2,y_2\right)$ is

  $m=\frac{y_2-y_1}{x_2-x_1}$

Equation of line having slope $m$ and passing through the point $P_1\left(x_1,y_1\right)$ is

  $y-y_1=m\left(x-x_1\right)$

Calculation:

Points are given as follows:

  $A\left(1,4\right)$ and $B\left(7,-2\right)$

Slope of the line segment is calculated as

  $\begin{array}{l}m=\frac{-2-4}{7-1}\\ m=\frac{-6}{6}\\ m=-1\end{array}$

Let us assume that slope of line perpendicular to above line is $m$

Since theProduct of the slope of perpendicular line is equal to $-1$

Thus,

  $mm=-1$

Substituting the values,

  $\begin{array}{l}m=\frac{-1}{-1}\\ m=1\end{array}$

Mid-point of the line segment joining above two points is calculated as

  $\left(x,y\right)=\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}$

Substituting the values,

  $\begin{array}{l}\left(x,y\right)=\frac{1+7}{2},\frac{4-2}{2}\\ \left(x,y\right)=\frac{8}{2},\frac{2}{2}\\ \left(x,y\right)=4,1\end{array}$

Thus, the perpendicular bisector passes through the point $\left(4,1\right)$

Now, equation of perpendicular bisector is

  $\begin{array}{l}y-1=1\left(x-4\right)\\ y-1=x-4\\ y=x-4+1\\ y=x-3\end{array}$

Conclusion:

Hence, equation of perpendicular bisector is $y=x-3$