CK12 - Geometry

• Use the PerpendicularBisectorTheoremto solveproblemsinvolvingthe circumcenterof triangles.

Introduction

In our last lessonwe examinedmidsegmentsof triangles.In this lessonwe will examineanotherconstruction
that can occurwithintriangles,calledperpendicularbisectors.

The perpendicularbisectorof a line segmentis the line that:

1. dividesthe line segmentinto two congruentsub-segments.


2. intersectsthe line segmentat a right angle.

Hereis an exampleof a perpendicularbisectorto line segment.

PerpendicularBisectorTheoremand its Converse

We can provethe followingpair of theoremsaboutperpendicularbisectors.

PerpendicularBisectorTheorem:If a point is on the perpendicularbisectorof a segment,

then it is equidistantfrom the endpointsof the segment.

Proof. Consider with perpendicularbisector with points and on line as follows:

We mustshowthat.

"1. Since is the perpendicularbisectorof , it followsthat and angles and

are congruentand are right angles.

By the SAS postulate,we have.

So by CPCTC(correspondingpartsof congruenttrianglesare congruent).

It turnsout that we can also provethe converseof this theorem.