Equal Chords of a Circle
Some Results on Equal Chords of a Circle1) In a circle equalchords are equidistant from the center.
If AB = CD then OM = OL
Converse of the above result is also true.
If OM = OL then AB = CD
2) Equalchords of congruent circles are equidistant from the corresponding centers.
If two circles are congruent and AB = CD then OL = PM.
Converse of the above result is also true.
If two circles are congruent and OL = PM then AB = CD.
3)In a circle equal chords subtend equal angles at the center.
In a circle, if AB = CD then ∠AOB = ∠COD
Converse of the above result is also true.
In a circle, if ∠AOB = ∠COD then AB = CD.
Some solved examples on the above result:
1) If two chords of a circle are equally inclined to the diameter through their point of intersection, prove that the chords are equal.
Given : ∠OAL = ∠OAM.
Prove that : AB = AC
Construction : Draw OL ⊥ AB and OM ⊥ AC


1)OL ⊥ AB and OM ⊥ AC  1) By construction 
2)∠OLA = ∠OMA  2)Each 90^{0} 
3) OA = OA  3) Reflexive (common ) 
4) ∠OAL = ∠OAM  4) Given 
5) ΔOLA = ΔOMA  5) AAS postulate 
6) OL = OM  6) CPCTC 
7) AB = CD  7) Chords are equidistant from center O 
2) If two equalchords of a circle intersect within the circle, prove that the line joining the point of intersection to the center makes equal angles with the chords.
Given : PQ = RS They intersect each other at point T.
Prove that : ∠OTV = ∠OTU
Construction: Draw perpendiculars OV and OU on these chords.


1) OV = OU  1) Equalchords of a circle are equidistant from the center 
2)∠OVT = ∠OUT  2)Each 90° 
3) OT = OT  3) Reflexive (common) 
4) ΔOVT ≅ ΔOUT  4) HL postulate or (RHS theorem) 
5) ∠OTV = ∠OTU  5) CPCTC 
Circles
• Circles
• Parts of Circle
• Arc and Chords
• Equal Chords of a Circle
• Arc and Angles
• Cyclic Quadrilaterals
• Tangent to Circle