Question

In the given figure $A,B,C$ are three points on the circle, $O$ is its centre. If $\angle AOB=2x^{\circ }$ and $\angle ACB=3x-{65}^{\circ }$, find the value of $x$.

Answer 1

According to the problem,

$\frac{1}{2}\angle AOB=\angle ACB$

we know,

[SINCE THE ANGLE SUBTENDED BY AN ARC AT THE CENTRE IS DOUBLE THE ANGLE SUBTENDED BY IT AT ANY POINT ON THE REMAINING PART OF THE CIRCLE]

hence,

$3x-65=\frac{1}{2}\times 2x$

$2x=65$

$x={32.5}^{\circ }$