What is the ratio of the perimeter of $OPRQ$ to the perimeter of $OPSQ$?

Question

Area of circle $O$ is $64\pi$. What is ratio of the perimeter of $OPRQ$ to that of $OPSQ$ ($\pi = 3$)?

Okay i have tried couple of things but seems its not working . Please suggest me proper solution of this example so that , i can solve similar questions.

Answer 1

arc length=$\dfrac {\theta}{360^\circ}\times2\pi r\;\;\;,$here $\theta\;$is angle made by arc on centre of circle. $$m\widehat {PRQ}=\dfrac{120}{360}\times 2\pi\cdot8$$ $$m\widehat {PRQ}=\dfrac{16\pi}{3}$$ $$ m\widehat{PRQ}=16$$

$C=48\;,$$ m\widehat{PSQ}=C-m\widehat{PRQ}\implies48-16=32$

so ratio=$\dfrac {m\widehat{OPRQ}}{m\widehat{OPSQ}}=\dfrac{16}{32}\implies\dfrac 12$

Answer 2

Hint. From the area you are given ($64\pi = \pi r^2$), determine the radius, that gives you $OP$ and $OQ$.

To get the arc $PQ$, think what proportion of the circumference $C=2\pi r$) must be covered by $PQ$ if it corresponds to an angle of 120 degrees (how many degrees are in the entire circle?)