Question

If two equal chords of a circle intersect within the circle, prove that the segments of one chord are equal to corresponding segments of the other chord.

Answer 1

Data : Circle with centre ‘O’, AB and CD chords are equal and intersect at E. 

To Prove: Segment EAD = Segment ACB.

Construction: Draw OP⊥CD and OQ⊥AB. 

Join OE. 

Proof: AQ = QB (∵ Chord drawn perpendicular CP 

= PD ,to centre bisects chord.). 

In ∆OPE and ∆OQE, 

∠OPE = ∠OQE = 90° 

(Construction) 

OP = OQ OE is common. 

∴ ∆OPE ≅ ∆OQE (RHS Theorem) 

QE = PE ……….. (i) 

But QB = PD 

∴ QB – QE = PD – PE 

EB = ED ……….. (ii) 

From (i) and (ii), 

AE = CE. 

CBD is minor segment formed by Chord CD. 

ADB is minor segment formed by Chord AB. 

∴ Minor segment CBD = Minor segment ADB 

Major segment CAD = Major segment ACB.