Question

circles chapter,pls help AB is the chord of a circle with centre O and AB
is produced to C such that BO = BC.
CO produced meets the circle at D.
If ∠ ACD = y and ∠ AOD = x, prove that x = 3y.

Answer 1

AB is a chord of a circle with centre O AB is produced to C such that BO = BC

CO is joined and produced to meet the circle at D

We shall prove x

∘

=3y

∘

We have

BC=OB

∠OCB=∠BOC=y

∘

[Angles opposite to equal sides are equal ]

∠OBA=∠BOC+∠OCB

[Ext angle of a △ is equal to the sum of the opposite interior angles ]

⇒∠OBC=y

∘

+y

∘

=2y

∘

OA=OB...[Radii of the same circle ]

∠OAB=∠OBA....[Angles opp. To equal sides of a △]

=2y

∘

∠AOD=∠OAC+∠OCA

=2y

∘

+y

∘

=3y

∘

[Exterior angle - Sum of opposite interrior angles]

⇒x

∘

=3y

∘