We are asked to find the equation of the perpendicular bisector of the segment ([PQ]), where: - (P(1,3)) - (Q(-1,2)).

Step 1: Midpoint of ([PQ])

The midpoint (M) has coordinates: $$M = \left(\frac{1 + (-1)}{2}, \frac{3 + 2}{2}\right) = (0, \tfrac{5}{2})$$

Step 2: Slope of (PQ)

The slope of the line (PQ) is: $$m_{PQ} = \frac{3 - 2}{1 - (-1)} = \frac{1}{2}$$

Step 3: Slope of the perpendicular bisector

The perpendicular slope is the negative reciprocal: $$m_{\perp} = -\frac{1}{m_{PQ}} = -2$$

Step 4: Equation of the perpendicular bisector

Equation through (M(0,)) with slope (-2): $$y - \frac{5}{2} = -2(x - 0)$$ $$y = -2x + \frac{5}{2}$$

✅ Final Answer

The equation of the perpendicular bisector of ([PQ]) is: $$\boxed{y = -2x + \tfrac{5}{2}}$$ ```