The perpendicular bisector of the segment passes through the midpoint of this segment. Thus, we will initially find the midpoint P:
[tex]P=\dfrac{(1,5)+(7,-1)}{2}=\dfrac{(8,4)}{2}=(4,2)[/tex]
Now, we will calculate the slope of the segment support line (r). After this, we will use the fact that the perpendicular bisector (p) is perpendicular to r:
[tex]m_r=\dfrac{\Delta y}{\Delta x}=\dfrac{5-(-1)}{1-7}=\dfrac{6}{-6}\iff m_r=-1[/tex]
[tex]p\perp r\Longrightarrow m_p\cdot m_r=-1\Longrightarrow m_p\cdot(-1)=-1\iff m_p=1[/tex]
We can calculate the equation of p by using its slope and its point P:
[tex]y-y_P=m_p(x-x_P)\\\\
y-2=1\cdot(x-4)\\\\
y-2=x-4\\\\
\boxed{p:~~y=x-2}[/tex]
