Question

$AB$ is the chord of the 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 $\angle ACD=y$ and $\angle AOD=x$,prove that $x=3y$.

Answer 1

Given,

$BC=OB$

$OB=OA\Rightarrow \angle ABO=\angle BAO$....(1)

$\angle AOD=x$

$\angle AOD=\angle BAO+\angle BCO$....(A)

From (1) $\to \angle BAO=2y$

$\angle BOC=\angle BCO=y$

From A

$\angle AOD=\angle BAO+\angle BCO$

$x=2y+y$

$x=3y$

Hence proved.