The perpendicular bisector of the line segment XY is the line that passes through the midpoint of XY, and is perpendicular to XY. First, find the midpoint of XY. If X= (x1,y1) and Y = (x2,y2) are the endpoints of a line segment, then the x- and y-coordinates of the midpoint of the segment XY are given by
(x1 + x2)/2 and (y1 + y2)/2,
respectively. Thus, in our case, the coordinates of the midpoint are
(5 + (-3))/2 = 1
(7 + 3)/2 = 5.
Thus, the midpoint of XY is (1,5). So we want to find the equation of the line that is perpendicular to XY and passes through (1,5). Now find the slope of the line segment XY. The slope is
m = (y2 - y1)/(x2 - x1)
m = (3 - 7)/(-3 - 5)
m = -4/(-8)
m = 1/2
Recall that if two (nonvertical) lines are perpendicular, there slopes are negative reciprocals. Therefore, since the slope of segment XY is 1/2, the slope of the perpendicular bisector will be m = -2. Finally, then, we can find the equation of the perpendicular bisector using the point-slope form for the equation of a line:
y - y1 = m(x - x1).
Using m = -2 and (x1,y1) = (1,5), we have
y - 5 = -2(x - 1)
Usually, we would put the final answer in slope-intercept form (y = mx + b form), so we simplify and solve the previous equation for y to obtain
y = -2x + 7
Hope that helps. Let me know if you need any further explanation.