CK12 - Geometry

Thereare severaltheoremsthat relateto chordsof a circlethat we will discussin the followingsections.

PerpendicularBisectorof a Chord

TheoremThe perpendicularbisectorof a chordis a diameter.

Proof

We will drawtwo chords, and suchthat is the perpendicularbisectorof.

We can see that for any point on chord.

The congruenceof the trianglescan be provenby the postulate:

and are right angles

This meansthat.

Any pointthat is equidistantfrom and lies along , by the perpendicularbisectortheorem.Since

the centerof the circleis one suchpoint,it mustlie along so is a diameter.

If is the midpointof then and are radii of the circleand is a diameterof the circle.

PerpendicularBisectorof a ChordBisectsInterceptedArc

TheoremThe perpendicularbisectorof a chordbisectsthe arc interceptedby the chord.

Proof