Question

Perpendicular Bisector Find an equation of the perpendicular bisector of the line segment joining the points$A\left(1,4\right)$ and$B\left(7,-2\right)$

Short answer

The perpendicular bisector is $x-y=3$.

Step-by-step solution

Step 1. Concept of Perpendicular bisector.

The line which cut a line perpendicularly at the mid-point is called the perpendicular bisector of that point and the slope of that bisectors are the negative reciprocal of the line.

Step 2. Determine the segment AB and mid-point of AB.

The points are $A\left(1,4\right),B\left(7,-2\right).$

$\begin{array}{c}AB=y-4=\frac{7-1}{-2-4}\left(x-1\right)\\ y-4=-1\left(x-1\right)\\ x+y=5\end{array}$

and the mid-point of$AB=\left(\frac{7+1}{2},\frac{-2+4}{2}\right)=\left(4,1\right).$

The slope of$AC=\frac{6-1}{21-1}=\frac{5}{20}=\frac{1}{4}=m_3$ (say).

Step 3. Determine the perpendicular bisector of AB.

The slope of$AB=-1$that’s imply the slope of perpendicular bisector isand the bisector 1 passes through the point$\left(4,1\right).$

Hence, the equation of perpendicular bisector are$\begin{array}{c}y-1=1\left(x-4\right)\\ x-y=3\end{array}$