Question

In the figure, O and O' are centres of two circles intersecting at B aand C. ACD is a straight line, find x.

Answer 1

In the figure, two circle with centers O and O' intersect each other at B and C.

ACD is a line, $\angle AOB={130}^{\circ }$

Arc AB subtends $\angle AOB$ at the centre O and $\angle ACB$ at the remaining part of the circle.

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

$=\frac{1}{2}\times {130}^{\circ }={65}^{\circ }$

But $\angle ACB+\angle BCD={180}^{\circ }\left(Linearpair\right)$
$\Rightarrow {65}^{\circ }+\angle BCD={180}^{\circ }$

$\Rightarrow BCD={180}^{\circ }-{65}^{\circ }={115}^{\circ }$

Now, are BD subtends reflex $\angle B{O}^{\prime }D$ at the centre and $\angle BCD$ at the remaining part of the circle

$\therefore \angle B{O}^{\prime }D=2\angle BCD=2\times {115}^{\circ }={230}^{\circ }$

But $\angle B{O}^{\prime }D+reflex\angle B{O}^{\prime }D={360}^{\circ }$ (Angles at a point)

$\Rightarrow x+{230}^{\circ }={360}^{\circ }$

$\Rightarrow x={360}^{\circ }-{230}^{\circ }={130}^{\circ }$

Hence $x={130}^{\circ }$