Perpendicular Bisector
Pronunciation: /ˌpɜr.pənˈdɪk.jə.lər ˈbaɪ.sɛk.tər/ Explain
Click on the blue points and drag them to change the figure.
Note that the perpendicular bisector is at right angles to the line segment.
 Manipulative 1  Perpendicular Bisector of a Line Segment Created with GeoGebra. 

A perpendicular bisector is a
line
that
bisects
a line segment and is
perpendicular
to the line segment. In manipulative 1, line segment
AB is the line segment being bisected.
The red line is the perpendicular bisector. Point
C is the midpoint of line segment
AB.

Properties of a Perpendicular Bisector
Click on the blue points and drag them to change the figure.
Is the any position of C where the two red line segments are not the same length?
 Manipulative 2  Perpendicular Bisector Property Created with GeoGebra. 

Each point on a perpendicular bisector are the same distance from the endpoints.
Since all points on a circle are the same distance from the center of the circle,
two circles of the same size can be used to find a perpendicular bisector.
In manipulative 2, point D is the
same distance from the center of each of the circles, meaning
AD ≡ BD. As the radius
AD changes, the points
D and
E are always on the perpendicular
bisector. Click on point D and drag it
to trace the perpendicular bisector.
The perpendicular bisectors of the sides of a triangle meet at the circumcenter of
the triangle.

Click on the blue points and drag them to change the figure
 Manipulative 3  Circumcircle of a Triangle Created with GeoGebra. 

The perpendicular bisectors of the sides of a triangle intersect at a point
called the circumcenter
of the triangle.

How to Construct a Perpendicular Bisector
Table 1 shows the steps to create a perpendicular bisector using a straight
edge and a compass. Click on the blue points in each of the manipulatives and
drag them to change the figure.
Step  Manipulative  Description  Justification 


Line segment AB is the line
segment to bisect. 
These are the criteria. 
1 

Draw a circle with center A and
radius AB. 
Euclid
Elements Book 1 Postulate 3: A circle can be draw with any center and
any radius.

2 

Draw a circle with center B and
radius BA. 
Euclid
Elements Book 1 Postulate 3: A circle can be draw with any center and
any radius.

3 

Mark the intersections of the circles as points
C and
D. 
An intersection is a point of concurrency. 
4 

Draw a line through points C and
D. This line is the
perpendicular bisector. 
Euclid. Elements
Book 1 Postulate 1. A line can be drawn from any point to any point.

5 

The intersection of line segment AB
and line segment CD is the midpoint
of line segment AB.

An intersection is a point of concurrency. 
Table 1: Constructing a perpendicular bisector. 
How to Find the Equation of a Perpendicular Bisector
Click on the blue points and drag them to change the figure.
Why is there different equations if the perpendicular bisector is vertical or horizontal?
 Manipulative 10  Equation of a Perpendicular Bisector of a Line Created with GeoGebra. 

The equation of a perpendicular bisector can be calculated for a given line
segment with end points (x_{1},
y_{1}) and
(x_{2}, y_{2}).
This demonstration will show how to calculate the equation of a perpendicular
bisector in point slope form.
Step  Equation  Description 
1 

First calculate the location of midpoint. 
2 

Calculate the slope of the line segment AB. 
3 

Calculate the slope of the perpendicular line using the slope of line
segment AB. 
4 

Put the slope of the perpendicular line and the coordinates of the
midpoint into and equation in point slope form. 
Table 2: Calculating the equation of a
perpendicular bisector 

References
 McAdams, David E.. All Math Words Dictionary, perpendicular bisector. 2nd Classroom edition 201501084799968. pg 137. Life is a Story Problem LLC. January 8, 2015. Buy the book
More Information
 Dendane, A. Perpendicular Bisector. Analyze Math. 3/12/2009. http://www.analyzemath.com/Geometry/PerpendicularBisector/PerpendicularBisector.html.
 Euclid of Alexandria. Elements. Clark University. 9/6/2018. https://mathcs.clarku.edu/~djoyce/elements/elements.html.
Cite this article as:
McAdams, David E. Perpendicular Bisector. 4/29/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/p/perpendicularbisector.html.
Image Credits
Revision History
4/29/2019: Changed equations and expressions to new format. (
McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (
McAdams, David E.)
12/1/2018: Removed broken links, updated license, implemented new markup, updated geogebra app. (
McAdams, David E.)
8/7/2018: Changed vocabulary links to WORDLINK format. (
McAdams, David E.)
11/15/2008: Initial version. (
McAdams, David E.)