Theorems on Chord
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In this section we will discuss Theorems on Chord.Theorems on Circles at both the Intermediate and Higher tier are difficult areas for students, which teachers often find equally difficult to deliver.Here, I have explained some Circle theorems based on circle so that student can understand them easily.
1) If two arcs of a circle are congruent, then corresponding chords are equal.
Given : PQ = BC
Prove that : ∠POQ = ∠BAC


1) BC = PQ  1) Given 
2) OP = AB  2) Radius of a circle 
3) OQ = AC  3) Radius of a circle 
4) ΔPOQ = ΔBAC  4) SSS Postulate 
5) ∠POQ = ∠BAC  5) CPCTC 
Given : PQ is a chord of a circle and OL ⊥ PQ.
Prove that : LP = LQ


1) OP = OQ  1) Radii of the same circle. 
2) OM = OM  2) Reflexive (common) 
3) ∠OMP = ∠OMQ  3) Each 90^{0} 
4) ΔPMO = ΔQMO  4) HL postulate(RHS) 
5) PM = MQ  5) CPCTC 
Given : AB = AC
Prove that : Center O lies on the bisector of ∠BAC


1) AB = AC  1) Given 
2) ∠BAM = ∠CAM  2) Given 
3) AM = AM  3) Reflexive (Common) 
4) ΔBAM = ΔCAM  4) SAS Postulate 
5) BM = CM and ∠BMA = ∠CMA  5) CPCTC 
6) ∠BMA + ∠CMA = 90^{0}  6) Linear pair angles 
7) AM = BM and ∠BMA = ∠CMA = 90^{0}  7) From (6) 
8) AM is the perpendicular bisector of BC  8) Definition of perpendicular bisector. 
9) AM passes through the center O.  9) Perpendicular bisector of a chord always passes through the center. 
Theorems on Circle
• Theorems on Chord
• Theorems on Chord and Subtended Angle
• Theorems on Arc and Angle
• Theorems on Cyclic Quadrilateral
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