First thing to do is find the coordinates of the mid-point of line segment . That is because the perpendicular bisector of the line segment intersects the line segment at the mid-point by definition of a perpendicular bisector.

midpoint formula:

xo = (x1 + x2)/2

yo = ( y1 + y2)/2

where x1 = 3, y1 = 8, x2 = 2, y2 = -5

xo = 2.5

yo = 1.5

the midpoint is (2.5, 1.5)

Next we need the slope, m, of the given line that contains the line segment:

m = (y1 - y2)/(x1 - x2)

m = (8 -(-5))/(3-2)

m = 13

The slope of a perpendicular is the negative reciprocal of the slope of the given line, that is:

m1 = -1/m

m1 = -1/13

the perpendicular bisector can be written as y = -x/13 + b

since the midpoint (2.5, 1.5) lies on y = -x/13 + b we have

1.5 = -2.5/13 + b

b = 1.5 + 2.5/13

b = 22/13

thus the equation of the perpendicular bisector of the line joining the points (3,8) and (2,-5) is

y = -x/13 + 22/13