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In the figure, $O$ and $O’$ are centres of two circles intersecting at $B$ and $C$. $ACD$ is a straight line, find $x$.
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Academic Mathematics NCERT Class 9

Given:

$O$ and $O’$ are centres of two circles intersecting at $B$ and $C$. $ACD$ is a straight line.

To do:

We have to find $x$.

Solution:

$\angle AOB = 130^o$

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

Therefore,

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

$= \frac{1}{2} \times 130^o$

$= 65^o$

$\angle ACB + \angle BCD = 180^o$ (Linear pair)

$65^o + \angle BCD = 180^o$

$\angle BCD = 180^o-65^o= 115^o$

arc $BD$ subtends reflex $\angle BO’D$ at the centre and $\angle BCD$ at the remaining part of the circle.

$\angle BO’D = 2\angle BCD$

$= 2 \times 115^o$

$= 230^o$

$\angle BO’D + reflex\ \angle BO’D = 360^o$ (Angles at a point)

$x + 230^o = 360^o$

$x = 360^o -230^o$

$x = 130^o$

Hence, $x = 130^o$.

Updated on: 10-Oct-2022

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