Perpendicular Bisector Calculator
Find the line that passes through the midpoint at 90°
Enter Segment Endpoints
Key Concept
The perpendicular bisector passes through the midpoint and has a slope that is the negative reciprocal of the original segment.
Visual Representation
Understanding Perpendicular Bisectors
A perpendicular bisector is a line that passes through the midpoint of a segment and is perpendicular (at 90°) to that segment. Every point on the perpendicular bisector is equidistant from the segment's endpoints.
How to Find a Perpendicular Bisector
- Find the midpoint: M = ((x₁+x₂)/2, (y₁+y₂)/2)
- Find the original slope: m = (y₂-y₁)/(x₂-x₁)
- Find the perpendicular slope: m_perp = -1/m (negative reciprocal)
- Write the equation: y - M_y = m_perp(x - M_x)
Special Cases
- Vertical segment: Perpendicular bisector is horizontal (y = constant)
- Horizontal segment: Perpendicular bisector is vertical (x = constant)
Connection to Triangles
In a triangle, the three perpendicular bisectors of the sides all meet at a single point called the circumcenter. This point is the center of the circumscribed circle (circumcircle) and is equidistant from all three vertices.
Frequently Asked Questions
What is a perpendicular bisector?
A perpendicular bisector is a line that divides a segment into two equal parts at a 90° angle. It passes through the midpoint of the segment.
What is the negative reciprocal?
If the original slope is m, the perpendicular slope is -1/m. For example, if m = 2, then the perpendicular slope is -1/2.
Why are all points on the bisector equidistant from the endpoints?
By definition, the perpendicular bisector creates two congruent right triangles from any point on it to the endpoints, proving equidistance.
How is this related to the circumcenter?
The circumcenter of a triangle is where all three perpendicular bisectors meet. It's equidistant from all three vertices.
What if the segment is vertical?
If the segment is vertical (undefined slope), the perpendicular bisector is horizontal with slope 0. The equation is y = midpoint y-coordinate.
What if the segment is horizontal?
If the segment is horizontal (slope = 0), the perpendicular bisector is vertical. The equation is x = midpoint x-coordinate.
How do I construct a perpendicular bisector with compass?
Set compass wider than half the segment. From each endpoint, draw arcs that intersect above and below. Connect the intersection points.
How is this different from an angle bisector?
A perpendicular bisector divides a segment at 90°. An angle bisector divides an angle into two equal parts. They're different concepts.