5. Two equal chords AB and CD of a circle withcentre O, intersect each other at point P insidethe circle. Prove that :AP CP(ii) BP# DP

5. Two equal chords AB and CD of a circle withcentre O, intersect each

1.	5. Two equal chords AB and CD of a circle withcentre O, intersect each other at point P insidethe circle. Prove that :AP CP(ii) BP# DP
Answer» Given:AB=CDto prove:PB=PDconst: draw OE and OQ perpendicular on AB and CD respectivelyproof:given AB and CD are two equal chords of the same circle.OE=OQ(equal chords of a circle are equidistant from the center.)now in triangle OEP and OQP,OE=OQOP=OP(common)angle OEP = OQP =90 degree,by constructiontherefore triangle OEP = OQP (RHS congruency)EP = QP (CPCT)also AE=EB=1/2 AB and CQ=QD=1/2CD (the line joining the center of the circle is perpendicular to the chord and bisects the chord.)Now AB = AC implies AE = EB= CQ=QD ....(1)therefore EP-BE =QP - BEEP - BE = QP - QD (FROM 1)BP = PDhence proved