Question

The perpendicular bisector to the segment between (-3,8) and (9,4)

Answer 1

What do you know about the perpendicular bisector ?

-- It's perpendicular to the line between these two points, so
its slope is the negative reciprocal of the slope of that line.

-- It goes through the midpoint between these two points.


The slope of the line between these two points is

   (the difference in their y-values) / (the difference in their x-values)

=                                          (4 - 8) / (9 - -3) 

=                                               -4  /  12  =       -1/3

So the slope of the perpendicular bisector is +3 .


The mid-point of the line between these two points is

       (the average of their x-values, the average of their y-values)

=    [                      1/2(-3+9)           ,             1/2(8+4)                    ]

=    [                       1/2(6)               ,              1/2(12)                      ]

=                                                  (3, 6)

There you go.  You know the slope of the perpendicular bisector,
and you know one point on it, so you're all set to write the equation
of the line in point-slope form.

Answer 2

You are looking for the midpoint, so:
x: between -3 and 9 is 3
y: between 8 and 4 is 6
therefore, the perpendicular bisector intersects at (3,6)