SAT Question 69b0d79d: Circles answer and explanation with Desmos solution

Choice B is correct. Because segments $OA$ and $OB$ are radii of the circle centered at point $O$, these segments have equal lengths. Therefore, triangle $AOB$ is an isosceles triangle, where angles $OAB$ and $OBA$ are congruent base angles of the triangle. It's given that angle $OAB$ measures $30^{\circ }$. Therefore, angle $OBA$ also measures $30^{\circ }$. Let $x^{\circ }$ represent the measure of angle $AOB$. Since the sum of the measures of the three angles of any triangle is $180^{\circ }$, it follows that $30^{\circ }+30^{\circ }+x^{\circ }=180^{\circ }$, or $60^{\circ }+x^{\circ }=180^{\circ }$. Subtracting $60^{\circ }$ from both sides of this equation yields $x^{\circ }=120^{\circ }$, or $\frac{2\pi }{3}$ radians. Therefore, the measure of angle $AOB$, and thus the measure of arc $AB$, is $\frac{2\pi }{3}$ radians. Since $\overline{OC}$ is a radius of the given circle and its length is $18$, the length of the radius of the circle is $18$. Therefore, the length of arc $AB$ can be calculated as $\frac{2\pi }{3}\times 18$, or $12\pi$.