Chapter 9.1, Problem 2P

To determine

To calculate: The equation of the perpendicular bisector.

Answer to Problem 2P

The required equation of the perpendicular bisector is $x-2y+4=0$ .

Explanation of Solution

Given information: The coordinates of points A and B are $\left(2,1\right)$ and $\left(-2,3\right)$ .

Formula used: $M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$ and $m_1\times m_2=-1$

Calculation: Let the perpendicular bisector is passing through point M.

Hence, the point M is a midpoint.

  $\begin{array}{cc}A& B\\ x_1=2& x_2=-2\\ y_1=1& y_2=3\end{array}$

  $\begin{array}{l}M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\\ M=\left(\frac{2-2}{2},\frac{1+3}{2}\right)\\ M=\left(0,2\right)\end{array}$

Now, calculate the slope of AB,

  $\begin{array}{l}{m}_1=\frac{x_2-x_1}{y_2-y_1}\\ m_1=\frac{-2-2}{3-1}\\ m_1=-2\end{array}$

Now, find the value of $m_2$ ,

  $\begin{array}{l}{m}_1\times m_2=-1\\ -2\times m_2=-1\\ m_2=\frac{-1}{-2}\\ \therefore m_2=\frac{1}{2}\end{array}$

The equation of perpendicular bisector is,

  $\begin{array}{l}y-y_1=m_2\left(x-x_1\right)\\ y-2=\frac{1}{2}\left(x-0\right)\\ y-2=\frac{x}{2}\\ \frac{x}{2}-y+2=0\\ x-2y+4=0\end{array}$

Thus, the required equation of perpendicular bisector is $x-2y+4=0$ .