1.

2.0 is the centre of the circle and AB is a chord. AC is the bisector of <OAB,<OAB=56a)Prove that OC and AB are parallel.b)Find <ABC and <OBE,B

Answer»

Given:OisthecentreandABisachord.ACisthebisectorof∠OABwhichis56°.

ToProve:OC∥ABthenToCalculate:∠ABCand∠OBE.

Solution:SinceACistheanglebisectorof∠OAB=56°

⇒∠OAC=∠CAB=562=28°

Now,fromarcBC⏜

⇒∠BOC=2∠CAB(Anglesubtendedbyanarconcentreisdoubletheanglesubtendedoncircumference)

⇒∠BOC=2×28°=56°

Now,OA=OB(Radiusofthecircle)so,△OABisisosceleswithOA=OBso,∠OAB=∠ABO=56°Now,since,∠ABO=∠BOC=56°⇒Alternateinterioranglesareequal⇒OC∥ABNow,in△OBCOB=OC(Radius)⇒∠OCB=∠OBC⇒∠BOC+2∠OBC=180°⇒56+2∠OBC=180°⇒2∠OBC=180°-56°⇒2∠OBC=124°⇒∠OBC=124°/2⇒∠OBC=62°Now,∠OBA+∠OBC+∠CBE=180°(anglesonastraightline)⇒56°+62°+∠CBE=180°⇒∠CBE=180°-56°-62°⇒∠CBE=62°Now,Requiredangles∠ABC=∠ABO+∠OBC=56°+62°=118°∠OBE=∠OBC+∠CBE=62°+62°=124


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