How to Find the Equation of a Perpendicular Bisector

You need the line that cuts a segment exactly in half and meets it at a right angle. That is the perpendicular bisector equation. Write it by finding the midpoint of the two endpoints, then calculating the negative reciprocal slope of the segment, and plugging both into the point-slope form of a line. Here is the exact method.

A perpendicular bisector is a line that passes through the midpoint of a line segment and is perpendicular to that segment. The midpoint is the average of the two endpoints, exactly one point. The line is the bisector: it splits the segment into two equal lengths and stands at a 90-degree angle to it. The term perpendicular bisector of a line segment names the whole line, not just the point where it crosses. Every point on that line is equidistant from the two endpoints, which is the property that makes it useful for locating the circumcenter of a triangle.

Follow these four steps in order. They work for any two distinct points in a Cartesian coordinate system, whether the coordinates are positive, negative, or fractional.

Step 1: Find the Midpoint

Use the midpoint formula: M = ((x₁ + x₂)/2, (y₁ + y₂)/2). This gives you the point the bisector must pass through. For a segment from (1, 3) to (5, 7), the midpoint is ((1+5)/2, (3+7)/2) = (3, 5).

Step 2: Find the Segment's Slope

The slope of the segment is (y₂ − y₁) / (x₂ − x₁). For the same segment, (7 − 3) / (5 − 1) = 4/4 = 1.

Step 3: Find the Negative Reciprocal Slope

The perpendicular slope is the negative reciprocal of the segment's slope. If the segment's slope is m, the perpendicular slope is −1/m. For a slope of 1, the negative reciprocal slope is −1. If the segment's slope is 2/3, the perpendicular slope is −3/2. If the segment is horizontal (slope 0), the perpendicular is vertical (undefined slope). If the segment is vertical, the perpendicular is horizontal.

Step 4: Write the Equation

Use the point-slope form: y − y₁ = m(x − x₁). Substitute the midpoint coordinates for (x₁, y₁) and the perpendicular slope for m. For midpoint (3, 5) and slope −1: y − 5 = −1(x − 3). Simplify to y = −x + 8. That is the equation of the perpendicular bisector.

Midpoint: ((2+6)/2, (1+5)/2) = (4, 3).
Segment slope: (5 − 1) / (6 − 2) = 4/4 = 1.
Perpendicular slope: −1/1 = −1.
Equation: y − 3 = −1(x − 4) → y = −x + 7.

Check: The line y = −x + 7 passes through (4, 3). At x = 2, y = 5; at x = 6, y = 1. The distance from (2, 1) to (4, 3) is √((4−2)² + (3−1)²) = √(4+4) = √8. The distance from (6, 5) to (4, 3) is also √8. The line is perpendicular, the product of the slopes (1)(−1) = −1. The equation is correct.

Horizontal segments have slope 0. The perpendicular bisector is vertical, with an equation of the form x = midpoint x-coordinate. For endpoints (1, 4) and (7, 4), the midpoint is (4, 4) and the bisector is x = 4.

Vertical segments have undefined slope. The perpendicular bisector is horizontal, with equation y = midpoint y-coordinate. For endpoints (3, 2) and (3, 8), the midpoint is (3, 5) and the bisector is y = 5.

The failure case: if the two endpoints are the same point, the segment has zero length and no unique perpendicular bisector. The problem is undefined, and calculators that return a line are wrong.

The circumcenter is the point equidistant from all three vertices of a triangle. It is the intersection of the perpendicular bisectors of any two sides. To find it, write the perpendicular bisector equations for two sides of the triangle using the four-step method, then solve the system of two linear equations. The solution is the circumcenter.

For a triangle with vertices A(0, 0), B(4, 0), and C(2, 3):
Side AB: midpoint (2, 0), slope 0, bisector x = 2.
Side BC: endpoints (4, 0) and (2, 3), midpoint (3, 1.5), slope (3−0)/(2−4) = 3/−2 = −1.5, perpendicular slope = 2/3. Equation: y − 1.5 = (2/3)(x − 3).
Solve x = 2 with y − 1.5 = (2/3)(−1) → y = 1.5 − 2/3 = 5/6. The circumcenter is (2, 5/6).

If the three points are collinear, they do not form a triangle and the circumcenter is undefined. The bisectors are parallel and never intersect.

The perpendicular bisector equation suits geometry students working with coordinate proofs, triangle centers, and partitioning problems. It also suits anyone who needs to find a line equidistant from two points, surveyors, mapmakers, and game developers placing objects symmetrically. It does not suit anyone looking for the center of a polygon or a set of three or more points. That is the centroid, a different calculation using a different formula. Nor does it suit someone who needs the midpoint of a time interval or a date range. The most common error: averaging three points and calling the result a midpoint. That gives a centroid, not a bisector, and the line through it is not perpendicular to anything.