Perpendicular Bisector

Perpendicular Bisector

The line perpendicular to a segment passing through the segment's midpoint. Note: The perpendicular bisectors of the sides of a triangle are concurrent, intersecting at the circumcenter.

Two lines intersecting: a diagonal segment with arrows on both ends, crossed by a perpendicular bisector line, both labeled.

See also

Circumcircle, centers of a triangle

Key Formula

y - y_m = -\frac{1}{m_{AB}}\,(x - x_m)

Where:

$(x_m, y_m)$ = The midpoint of segment AB, calculated as ((x_A + x_B)/2, (y_A + y_B)/2)
$m_{AB}$ = The slope of segment AB, calculated as (y_B - y_A)/(x_B - x_A)
$-1/m_{AB}$ = The negative reciprocal of the segment's slope, which gives the slope of the perpendicular bisector

Worked Example

Problem: Find the equation of the perpendicular bisector of the segment with endpoints A(2, 4) and B(8, 10).

Step 1: Find the midpoint of AB by averaging the coordinates.

M = \left(\frac{2+8}{2},\, \frac{4+10}{2}\right) = (5,\, 7)

Step 2: Calculate the slope of segment AB.

m_{AB} = \frac{10 - 4}{8 - 2} = \frac{6}{6} = 1

Step 3: Find the slope of the perpendicular bisector by taking the negative reciprocal.

m_{\perp} = -\frac{1}{1} = -1

Step 4: Write the equation using point-slope form with the midpoint and perpendicular slope.

y - 7 = -1(x - 5)

Step 5: Simplify to slope-intercept form.

y = -x + 12

Answer: The perpendicular bisector of segment AB is the line y = −x + 12.

Another Example

This example shows the edge case where the original segment is vertical, making its slope undefined. The perpendicular bisector becomes a horizontal line rather than requiring the standard negative reciprocal calculation.

Problem: Find the equation of the perpendicular bisector of the segment with endpoints P(1, 3) and Q(1, 9).

Step 1: Find the midpoint of PQ.

M = \left(\frac{1+1}{2},\, \frac{3+9}{2}\right) = (1,\, 6)

Step 2: Calculate the slope of segment PQ. Since both x-coordinates are equal, the segment is vertical.

m_{PQ} = \frac{9-3}{1-1} = \frac{6}{0} \;\Rightarrow\; \text{undefined (vertical line)}

Step 3: A line perpendicular to a vertical line is horizontal, so the perpendicular bisector has slope 0.

m_{\perp} = 0

Step 4: A horizontal line through the midpoint (1, 6) has the equation:

y = 6

Answer: The perpendicular bisector is the horizontal line y = 6.

Frequently Asked Questions

What is the difference between a perpendicular bisector and a median of a triangle?

A perpendicular bisector of a side passes through that side's midpoint at 90° and does not necessarily go through any vertex. A median connects a vertex to the midpoint of the opposite side and is generally not perpendicular to that side. The perpendicular bisectors meet at the circumcenter, while the medians meet at the centroid.

Why is every point on a perpendicular bisector equidistant from the segment's endpoints?

Take any point P on the perpendicular bisector of segment AB. The bisector passes through midpoint M, creating two right triangles, PMA and PMB. These triangles share side PM, have equal sides MA = MB (since M is the midpoint), and both contain a right angle. By the SAS congruence theorem (or the Pythagorean theorem), PA = PB. This equidistance property is the defining characteristic of a perpendicular bisector.

How do you find the perpendicular bisector of a segment on a coordinate plane?

First, compute the midpoint of the segment by averaging the x- and y-coordinates. Then find the slope of the segment and take its negative reciprocal to get the perpendicular slope. Finally, use point-slope form with the midpoint and the perpendicular slope. If the segment is vertical, the bisector is horizontal, and vice versa.

Perpendicular Bisector vs. Angle Bisector

	Perpendicular Bisector	Angle Bisector
What it bisects	A line segment, splitting it into two equal lengths	An angle, splitting it into two equal angles
Orientation	Always perpendicular (90°) to the segment	Lies within the interior of the angle; not necessarily perpendicular to anything
Equidistance property	Points on it are equidistant from the segment's two endpoints	Points on it are equidistant from the angle's two sides
In a triangle, they meet at	The circumcenter (center of the circumscribed circle)	The incenter (center of the inscribed circle)

Why It Matters

Perpendicular bisectors appear frequently in geometry courses, especially in proofs involving congruence and in constructions with compass and straightedge. In triangle geometry, the three perpendicular bisectors of the sides meet at the circumcenter, which is the center of the circle passing through all three vertices—a result used in coordinate geometry problems and in real-world applications like determining the location equidistant from three given points. You will also encounter perpendicular bisectors in analytic geometry when finding the equation of a circle given three points on it.

Common Mistakes

Mistake: Using the slope of the segment as the slope of the perpendicular bisector instead of the negative reciprocal.

Correction: If the segment has slope m, the perpendicular bisector has slope −1/m. Remember: perpendicular lines have slopes that are negative reciprocals of each other (their product is −1).

Mistake: Confusing the perpendicular bisector of a triangle's side with the altitude from the opposite vertex.

Correction: An altitude drops from a vertex and is perpendicular to the opposite side (or its extension), but does not necessarily pass through that side's midpoint. A perpendicular bisector always passes through the midpoint but does not necessarily go through a vertex. They coincide only for isosceles or equilateral triangles on the relevant side.

Related Terms

Midpoint — The point through which the perpendicular bisector passes
Perpendicular — The 90° relationship the bisector has to the segment
Line Segment — The segment being bisected
Circumcenter — Where the three perpendicular bisectors of a triangle meet
Circumcircle — Circle centered at the circumcenter through all three vertices
Concurrent — Property of the three perpendicular bisectors meeting at one point
Triangle — Shape whose sides have perpendicular bisectors meeting at circumcenter
Centers of a Triangle — The circumcenter is one of the notable triangle centers