Circles SAT question

Choice B is correct. Since $\overline{\mathrm{PR}}$ and $\overline{\mathrm{QS}}$ are diameters of the circle shown, $\overline{\mathrm{OS}}$, $\overline{\mathrm{OR}}$, $\overline{\mathrm{OP}}$, and $\overline{\mathrm{OQ}}$ are radii of the circle and are therefore congruent. Since $\angle SOP$ and $\angle ROQ$ are vertical angles, they are congruent. Therefore, arc $PS$ and arc $QR$ are formed by congruent radii and have the same angle measure, so they are congruent arcs. Similarly, $\angle SOR$ and $\angle POQ$ are vertical angles, so they are congruent. Therefore, arc $SR$ and arc $PQ$ are formed by congruent radii and have the same angle measure, so they are congruent arcs. Let $x$ represent the length of arc $SR$. Since arc $SR$ and arc $PQ$ are congruent arcs, the length of arc $PQ$ can also be represented by $x$. It’s given that the length of arc $PS$ is twice the length of arc $PQ$. Therefore, the length of arc $PS$ can be represented by the expression $2x$. Since arc $PS$ and arc $QR$ are congruent arcs, the length of arc $QR$ can also be represented by $2x$. This gives the expression $x+x+2x+2x$. Since it's given that the circumference is $144\pi$, the expression $x+x+2x+2x$ is equal to $144\pi$. Thus $x+x+2x+2x=144\pi$, or $6x=144\pi$. Dividing both sides of this equation by $6$ yields $x=24\pi$. Therefore, the length of arc $QR$ is $2(24\pi )$, or $48\pi$.

Choice A is incorrect. This is the length of arc $PQ$, not arc $QR$.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.