Question 1108685
The perpendicular bisector of a segment:
1) bisects the segment (meaning it passes through the midpoint of the segment)
2) is perpendicular to the segment (meaning that when you multiply the slopes of the perpendicular bisector and the line containing the segment, the result is {{{-1}}} ).
 
The coordinates of the midpoint {{{M(x[M],y[M])}}}
are the average of the coordinates of the end points, so
{{{x[M]=(7+(-3))/2=4/2=2}}}
{{{y[M]=((-1)+5)/2=4/2=2}}}
So, the midpoint is {{{M(2,2)}}}
 
The slope of the line containing segment PQ is
{{{m[PQ]=(-1-5)/(7-(-3))=(-6)/(7+3)=(-6)/10=-3/5}}} .
The slope of a perpendicular line is
{{{m=(-1)}}}{{{"/"}}}{{{(-3/5)=(-1)(-5/3)=5/3}}}
 
So, the perpendicular bisector of PQ is the line
with slope {{{m=5/3}}} that passe through {{{M(2,2)}}} .
In point-slope form, based on point {{{M}}} ,
the equation of that line is
{{{highlight(y-2=(5/3)(x-2))}}} .
That is just one of the infinitely many different forms of the equation of that line.
Solving for {{{y}}} ,
we find the one and only slope-intercept form of the equation of that line.
{{{y-2=(5/3)(x-2)}}}
{{{y-2=(5/3)x-10/3)}}}
{{{y=(5/3)x-10/3+2}}}
{{{highlight(y=(5/3)x-4/3)}}}