SAT Question 95ba2d09: Circles answer and explanation with Desmos solution

Choice C is correct. Because points $T$, $O$, and $P$ all lie on the $x$-axis, they form a line. Since the angles on a line add up to $180^{\circ }$, and it's given that $\angle POQ$ and $\angle ROT$ each measure $30^{\circ }$, it follows that the measure of $\angle QOR$ is $180^{\circ }-30^{\circ }-30^{\circ }=120^{\circ }$. Since the arc of a complete circle is $360^{\circ }$ or $2\pi$ radians, a proportion can be set up to convert the measure of $\angle QOR$ from degrees to radians: $\frac{360^{\circ }}{2\pi \text{ radians}}=\frac{120^{\circ }}{x\text{ radians}}$, where $x$ is the radian measure of $\angle QOR$. Multiplying each side of the proportion by $2\pi x$ gives $360x=240\pi$. Solving for $x$ gives $x=\frac{240\pi }{360}$, or $\frac{2}{3}\pi$.