Question

AB is the chord of a circle with centre O AB is produced to C such that BC=OB CO is joined and produced to meet the circle in D IF $\angle ACD=y^{\circ }and\angle AOD=x^{\circ }$
prove that $x^{\circ }=3y^{\circ }$

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 $x^{\circ }=3y^{\circ }$
We have
$BC=OB$

$\angle OCB=\angle BOC=y^{\circ }$
[Angles opposite to equal sides are equal ]

$\angle OBA=\angle BOC+\angle OCB$
[Ext angle of a $△$ is equal to the sum of the opposite interior angles ]

$\Rightarrow \angle OBC=y^{\circ }+y^{\circ }=2y^{\circ }$

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

$\angle OAB=\angle OBA$....[Angles opp. To equal sides of a $△$]

$=2y^{\circ }$

$\angle AOD=\angle OAC+\angle OCA$

$=2y^{\circ }+y^{\circ }$

$=3y^{\circ }$ [Exterior angle - Sum of opposite interrior angles]

$\Rightarrow x^{\circ }=3y^{\circ }$