We can see that becauseof the postulate.and are right angles.This meansthat.This completesthe proof.CongruentChordsEquidistantfromCenter
TheoremCongruentchordsin the samecircleare equidistantfrom the centerof the circle.
First,recallthat the definitionof distancefrom a pointto a line is the lengthof the perpendicularsegment
drawnto the line from the point. and are thesedistances,and we mustprovethat they are equal.
Proof.
by the SSS Postulate.(given)(radii)(radii)Sincethe trianglesare congruent,their correspondingaltitudes and mustalso be congruent:
.Therefore, and are equidistantfrom the center.Converseof CongruentChordsTheorem
TheoremTwo chordsequidistantfrom the centerof a circleare congruent.
This proofis left as a homeworkexercise.
Next,we will solvea few examplesthat applythe theoremswe discussed.
Example 1