To prove a line is a perpendicular bisector, you need to show two things: the line passes through the midpoint of a segment, and it meets that segment at a 90-degree angle. How you demonstrate those two facts depends on whether you’re working with coordinates, a formal two-column proof, or a construction. Here are the main approaches, each broken down step by step.
The Two Conditions You Must Prove
A perpendicular bisector of a segment AB is a line that is perpendicular to AB and passes through its midpoint. Every proof method, regardless of format, boils down to establishing both of these conditions. Showing only one isn’t enough. A line can pass through the midpoint at a slant (a bisector, but not perpendicular), or it can cross the segment at 90 degrees without hitting the midpoint (perpendicular, but not a bisector). You need both.
Proving It With Coordinates
This is the most common method in algebra-based geometry courses. You’re given (or you choose) coordinates for the endpoints of a segment, and you find the equation of the perpendicular bisector algebraically. There are four steps.
Step 1: Find the Midpoint
Use the midpoint formula. For endpoints (x₁, y₁) and (x₂, y₂), the midpoint is:
M = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)
For example, if A = (−4, 7) and B = (6, −3), the midpoint is ((−4 + 6) / 2, (7 + (−3)) / 2) = (1, 2). This midpoint is where your perpendicular bisector must pass through.
Step 2: Find the Slope of the Segment
Calculate the slope of AB using the standard slope formula: m = (y₂ − y₁) / (x₂ − x₁). With the same example, that’s (−3 − 7) / (6 − (−4)) = −10 / 10 = −1.
Step 3: Find the Perpendicular Slope
Two lines are perpendicular when their slopes are negative reciprocals of each other. That means you flip the fraction and change the sign. Mathematically, if the original slope is m₁, the perpendicular slope m₂ satisfies m₁ × m₂ = −1. Since the segment’s slope is −1, the perpendicular slope is 1 (because −1 × 1 = −1). If the segment’s slope were 2/3, the perpendicular slope would be −3/2.
Step 4: Write the Equation
Plug the midpoint and perpendicular slope into the point-slope form of a line: y − y₁ = m(x − x₁). Using midpoint (1, 2) and slope 1:
y − 2 = 1(x − 1), which simplifies to y = x + 1.
That equation is your perpendicular bisector. If you’re asked to prove a given line is the perpendicular bisector, you’d show that the line passes through the midpoint (by substituting the midpoint into the equation) and that its slope is the negative reciprocal of the segment’s slope.
Proving It With Triangle Congruence
In a formal Euclidean geometry proof, typically written in two-column format, you prove a perpendicular bisector by showing that two triangles on either side of the line are congruent. This approach works when you’re given a geometric diagram rather than coordinates.
The basic strategy: if a line splits a segment into two equal halves and you can prove the triangles formed on either side are congruent, you can conclude the line is a perpendicular bisector. The congruence theorems you’ll typically use are SAS (Side-Angle-Side) or ASA (Angle-Side-Angle), depending on what information you’re given. Once the triangles are congruent, corresponding parts of congruent triangles are equal (often abbreviated CPCTC), which lets you establish both the equal lengths (bisection) and the right angle (perpendicularity).
For example, suppose you have segment AC with point B on it and a line BD. If you can show triangles ABD and CBD are congruent using SAS, then CPCTC tells you AB = CB (so the segment is bisected) and that the angles at B are equal. Since those two equal angles are supplementary (they add to 180°), each must be 90°, proving perpendicularity.
Proving It With Equidistance
There’s a powerful shortcut that lets you skip triangle congruence entirely. The Perpendicular Bisector Theorem and its converse together state that the perpendicular bisector of a segment is exactly the set of points equidistant from both endpoints. This breaks into two usable statements:
- Theorem: If a point is on the perpendicular bisector of a segment, it is equidistant from both endpoints.
- Converse: If a point is equidistant from both endpoints of a segment, it lies on the perpendicular bisector.
The converse is especially useful for proofs. If you can identify two points that are each equidistant from the endpoints of your segment, those two points determine the perpendicular bisector. You simply show each point is the same distance from both endpoints, and you’re done. For instance, if points S and O are both equidistant from the endpoints R and H of a segment, then line SO is the perpendicular bisector of segment RH. This method works well when the diagram gives you information about equal distances (like radii of circles or marked congruent segments) but doesn’t directly give you angle measures or midpoints.
Proving It With a Compass Construction
A compass-and-straightedge construction is itself a proof, because each step relies on geometric properties that guarantee the result. Here’s the standard construction:
- Place the compass point on one endpoint of the segment. Set the radius to more than half the segment’s length.
- Draw an arc above and below the segment.
- Without changing the radius, place the compass on the other endpoint and draw arcs that intersect the first pair.
- Draw a line through the two intersection points.
Why this works: because you used the same compass radius from both endpoints, each intersection point is equidistant from both endpoints. By the converse of the Perpendicular Bisector Theorem, those two equidistant points determine the perpendicular bisector. If a test or assignment asks you to justify the construction, that equidistance argument is your proof.
Choosing the Right Method
The method you use depends on what you’re given. If the problem provides coordinates, use the algebraic approach with midpoint and slope calculations. If you’re working with a geometric diagram and need a two-column or paragraph proof, triangle congruence or the equidistance theorem will be your tools. If the problem involves a construction, the compass method with an equidistance justification is the way to go.
Regardless of method, the core logic is always the same. You are showing that a line does two things simultaneously: it cuts a segment in half, and it does so at a right angle. Every calculation, congruence statement, or construction step you write should point back to one of those two requirements.