The circle has circumference [tex]18\pi\,\mathrm{in}[/tex]. If [tex]m\angle POQ=\theta[/tex], then
[tex]\dfrac{6\pi\,\rm in}{18\pi\,\rm in}=\dfrac\theta{2\pi\,\rm rad}\implies\theta=\dfrac{2\pi}3\,\mathrm{rad}[/tex]
Answer: [tex]\dfrac{2\pi}{3}\text{ radians}[/tex]
Step-by-step explanation:
We know that the formula to calculate the length of arc having central angle 'x' is given by :-
[tex]l=x r[/tex], where r is radius of the circle.
Given : In circle O, radius OQ = [tex]l=9\text{ inches}[/tex]
The measure of arc PQ =[tex]6\pi\text{ inches.}[/tex]
The measure of central angle POQ ( in radians ) is given by :-
[tex]x=\dfrac{l}{r}\\\\\Rightarrow\ x=\dfrac{6\pi}{9}}\\\\\Rightarrow\ x=\dfrac{2\pi}{3}[/tex]
Hence, the measure of central angle POQ =[tex]\dfrac{2\pi}{3}\text{ radians}[/tex]
