1. The problem is to understand what a perpendicular bisector is. 2. A perpendicular bisector of a line segment is a line that divides the segment into two equal parts at a 90-degree angle. 3. The formula or rule is: If you have a segment with endpoints $A(x_1,y_1)$ and $B(x_2,y_2)$, the midpoint $M$ is given by $$M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$$ 4. The slope of the segment $AB$ is $$m=\frac{y_2-y_1}{x_2-x_1}$$ 5. The slope of the perpendicular bisector is the negative reciprocal of $m$, which is $$m_{\perp}=-\frac{1}{m}$$ 6. The perpendicular bisector passes through the midpoint $M$ and has slope $m_{\perp}$. 7. So its equation can be written as $$y - y_M = m_{\perp}(x - x_M)$$ where $(x_M,y_M)$ are the coordinates of the midpoint. 8. In simple terms, the perpendicular bisector cuts the segment exactly in half and forms a right angle with it. This concept is important in geometry for constructing triangles, finding circumcenters, and solving problems involving distances and symmetry.