Question

Determine the angle $\angle AOC,$ if $\angle CBA={25}^o$ and O is the centre of the circle.

• A. ${30}^o$
• B. ${40}^o$
• C. ${50}^o$
• D. ${60}^o$

Answer 1

The correct option is C ${50}^o$

The angle subtended by the diameter to any point on the circle is equal to ${90}^o.$

From the given figure,


$\angle ACB$ is the angle subtended by the diameter on the circle.
$\therefore \angle ACB={90}^o$

$\angle CAB={180}^o-(\angle CBA+{90}^o)$
$\Rightarrow \angle CAB={180}^o-{25}^o-{90}^o={65}^o$

For $△ACO,$ the sides AO and CO are radii of the circle.
Hence, $\angle CAO=\angle ACO$
Also, $\angle CAO+\angle ACO+\angle AOC={180}^o$
$\Rightarrow 2\angle CAO+\angle AOC={180}^o$
$\Rightarrow \angle AOC={180}^o-2\angle CAO$
$\Rightarrow \angle AOC={180}^o-2\times {65}^o={50}^o$