Enter two endpoints to find the perpendicular bisector line equation
Point A x₁
Point A y₁
Point B x₂
Point B y₂
Result
Midpoint M
Perpendicular Bisector
Detailed Derivation
Perpendicular Bisector Formula
Midpoint M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
Slope m = (y₂ − y₁) / (x₂ − x₁)
Perpendicular slope m⊥ = −1/m
y − y₀ = m⊥(x − x₀)
A perpendicular bisector is a line that cuts a line segment into two equal parts at a 90-degree angle. It is the locus of all points equidistant from the segment endpoints.
⚠If points A and B are vertical (x₁ = x₂), the perpendicular bisector is horizontal y = yₘ. If they are horizontal (y₁ = y₂), it is vertical x = xₘ.
What Is a Perpendicular Bisector?
A perpendicular bisector is a line that intersects a segment at its midpoint and forms a 90-degree angle. Every point on this line is exactly the same distance from both endpoints of the segment.
Midpoint
The midpoint M is the average of the endpoints: ((x₁+x₂)/2, (y₁+y₂)/2). The bisector always passes through M.
Perpendicular Slope
The new slope is the negative reciprocal: m⊥ = −1/m. The product of the two slopes equals −1.
Point-Slope Form
Using M(x₀,y₀) and m⊥: y − y₀ = m⊥(x − x₀). Convert to y = mx + b by distributing.
Geometric Property
Any point on the perpendicular bisector is equidistant from A and B. This is used in circumcenter construction.
Teaching Example: Points A(3,2) and B(7,6). Midpoint M = (5,4). Slope m = (6−2)/(7−3) = 1. Perpendicular slope = −1. Equation: y − 4 = −1(x − 5) → y = −x + 9.
A perpendicular bisector is a line that passes through the midpoint of a segment at a right angle. Every point on it is equidistant from the two endpoints.
How to find the perpendicular bisector equation?▼
Compute the midpoint, find the segment slope, take the negative reciprocal, then apply the point-slope formula with the midpoint.
What if the segment is vertical?▼
If the segment is vertical (x constant), the perpendicular bisector is a horizontal line through the midpoint: y = yₘ.
Why is the perpendicular bisector important?▼
It is essential for finding the circumcenter of a triangle, constructing perpendiculars, and solving geometry problems involving equal distances from two points.
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