The equation to the perpendicular bisector of the line segment joining $\left(1,2\right) and \left(3,4\right)$ is

Let the points $A\left(1 ,2\right) and B\left(3 ,4\right)$

$mid point of AB=\left(\frac{1+3}{2} , \frac{2+4}{2}\right)=\left(2 , 3\right)$

$slope of AB =\frac{4-2}{3-1}=1$

$\Rightarrow slope of perpendicular bisector =\frac{-1}{slope of AB}=\frac{-1}{1}=-1$

$$\begin{gathered} let m=-1 and \left(2 ,3 \right)=\left(x_1 , y_1\right) \\ \therefore Required equation is y-y_1=m\left(x-x_1\right) \\ \Rightarrow y-3=-1\left(x-2\right) \\ \Rightarrow y-3=-x+2 \\ \Rightarrow x+y-5=0 \end{gathered}$$