The given points are (-5,3), and (7,2).
First, we have to find the slope of the line that passes through the given points.
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]Replacing the given points, we have
[tex]m=\frac{2-3}{7-(-5)}=\frac{-1}{7+5}=-\frac{1}{12}[/tex]Since we have to find a perpendicular bisector to the line that passe through the given points, we have to find the perpendicular slope to -1/12.
[tex]m_1\cdot m=-1[/tex]Replacing the slope, we have
[tex]\begin{gathered} m_1\cdot(-\frac{1}{12})=-1 \\ m_1=12 \end{gathered}[/tex]So, the perpendicular bisector has a slope of 12.
Additionally, a perpendicular bisect passes through the midpoint between (-5,3) and (7,2), so let's find it
[tex]\begin{gathered} M=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}) \\ M=(\frac{-5+7}{2},\frac{3+2}{2}) \\ M=(\frac{2}{2},\frac{5}{2}) \\ M=(1,\frac{5}{2}) \end{gathered}[/tex]Now, we use this midpoint, the slope, and the point-slope formula to find the equation of the perpendicular bisector
[tex]\begin{gathered} y-y_1=m(x-x_1) \\ y-\frac{5}{2}=12(x-1) \\ y=12x-12+\frac{5}{2} \\ y=12x-\frac{24+5}{2} \\ y=12x-\frac{29}{2} \end{gathered}[/tex]
