Perpendicular Bisector of a Segment

Statement

The perpendicular bisector of a segment is the line that is perpendicular to the segment and passes through its midpoint. If the segment has slope \(m\), then the perpendicular slope is:

\[ m_\perp = -\frac{1}{m} \]

The line passes through the midpoint:

\[ \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right) \]

Why it’s true

• The midpoint formula gives the exact halfway point between two endpoints of a segment.
• The negative reciprocal slope ensures the new line is perpendicular to the original segment.
• Together, they uniquely define the perpendicular bisector.

Recipe (how to use it)

1. Find the slope of the segment: \(m=\frac{y_2-y_1}{x_2-x_1}\).
2. Compute the perpendicular slope: \(m_\perp=-1/m\).
3. Find the midpoint: \(\big(\tfrac{x_1+x_2}{2}, \tfrac{y_1+y_2}{2}\big)\).
4. Write the line equation using point–slope form: \(y-y_0=m_\perp(x-x_0)\).

Spotting it

Perpendicular bisectors often appear in coordinate geometry problems, circle theorems, and questions involving right angles or symmetry.

Common pairings

• Midpoint and slope calculations.
• Equations of lines.
• Circle center-finding problems.

Mini examples

1. Given: Points (0,0) and (4,4).
   Answer: Midpoint (2,2), slope = 1, perpendicular slope = -1, equation: \(y-2=-1(x-2)\).
2. Given: Points (2,3) and (6,3).
   Answer: Midpoint (4,3), slope = 0, perpendicular slope = undefined, bisector is vertical line \(x=4\).

Pitfalls

• Forgetting negative reciprocal for perpendicular slope.
• Mixing midpoint with slope formula.
• Special cases: horizontal lines ⟶ perpendicular is vertical (undefined slope), vertical lines ⟶ perpendicular is horizontal (slope 0).

Exam strategy

• Always compute midpoint first—it’s the anchor point for the bisector.
• Check slope carefully for special cases (0 or undefined).
• Use point–slope form to avoid mistakes when substituting values.

Summary

The perpendicular bisector of a line segment has slope \(m_\perp=-1/m\) and passes through the midpoint \(\big(\tfrac{x_1+x_2}{2}, \tfrac{y_1+y_2}{2}\big)\). It is the line that divides the segment into two equal parts at right angles.