To find the equation of the perpendicular bisector of PQ, we need to first find the midpoint of PQ, and then determine the slope of the line perpendicular to PQ that passes through this midpoint.
The midpoint of PQ is given by the formula:
((x1 + x2) / 2, (y1 + y2) / 2)
where (x1, y1) = (5, -4) and (x2, y2) = (-1, -2)
So, the midpoint of PQ is:
((5 - 1) / 2, (-4 - 2) / 2) = (2, -3)
Now, the slope of the line PQ can be found using the formula:
slope = (y2 - y1) / (x2 - x1)
where (x1, y1) = (5, -4) and (x2, y2) = (-1, -2)
So, the slope of PQ is:
(-2 - (-4)) / (-1 - 5) = 2/3
Since the perpendicular bisector of PQ will have a slope that is the negative reciprocal of the slope of PQ, the slope of the perpendicular bisector is:
-1 / (2/3) = -3/2
Finally, we can use the point-slope form of the equation of a line to find the equation of the perpendicular bisector. We know that the line passes through the point (2, -3), and has a slope of -3/2. So, the equation of the line is:
y - (-3) = (-3/2)(x - 2)
Simplifying this equation, we get:
y + 3 = (-3/2)x + 3
y = (-3/2)x + 0
Hence, the equation of the perpendicular bisector of PQ is y = (-3/2)x.