Question

If two equal chords of a circle intersect within the circle, prove that the line joining the point of intersection to the centre makes equal angles with the chords

Answer 1

Data: O is the centre of Circle. 

Chord AB = Chord CD. 

These chords intersects at E. 

OE is joined. 

To Prove: ∠OEP = ∠OEQ 

Construction: Draw OP⊥CD and OQ⊥AB. 

Proof: In ∆OPE and ∆OQE, 

∠OPE = ∠OQE = 90° 

(construction) OE is common. 

OP = OQ (Equal chords are equidistant) 

∴ ∆OPE ≅ ∆OEQ (RHS postulate) 

∴ ∠OEP = ∠OEQ.