A cyclic quadrilateral ABCD is drawn in a circle with centre O. A and C are joined to O. If $\angle A B C=2 p$ and $\angle A D C=3 p$, what is the measure (in degrees) of the $\angle A O C$ reflex?
Answer & explanation
Correct answer: option 4
\(\angle\)ABC + \(\angle\)ADC = \({180}^\circ\)
⇒ 2p + 3p = \({180}^\circ\)
⇒ 5p = \({180}^\circ\)
⇒ p = \({180}^\circ\)/5 = \({36}^\circ\)
So, \(\angle\)ABC = 2 x \({36}^\circ\) = \({72}^\circ\) and \(\angle\)ADC = 3 x \({36}^\circ\) = \({108}^\circ\)
Then \(\angle\)AOC = 2 x \({72}^\circ\) = \({144}^\circ\) [as, \(\angle\)ABC = \({72}^\circ\)] (Concept used)
Another side of \(\angle\)AOC = \({360}^\circ\) - \({144}^\circ\) = \({216}^\circ\)
Therefore, \(\angle\)AOC = \({216}^\circ\)