AB and CD are the chords of a circle whose center is O. They intersect each other at P. If PO is the bisector of ∠APD, prove that AB = CD. [2 MARKS]
Concept: 1 Mark
Application: 1 Mark
Given :AB and CD are the chords of a circle whose center is O.
They intersect each other at P.
PO is the bisector of ∠APD
To prove: AB = CD
Construction: Draw OR⊥AB and OQ⊥CD.
Proof: In ΔOPR and ΔOPQ
∠OPR=∠OPQ (Given)
OP=OP (Common)
∠ORP=∠OQP (Each =90∘ by construction)
∴ΔOPR=ΔOPQ (AAS congruency)
∴OR=OQ (C.P.C.T)
∴AB=CD (∵ chord of a circle which are equidistant from the center are equal).