OG详解-OG9 数学1 Q12

Choice B is correct. Since  PR​  and QS​ are diameters of the circle shown, OS​, OR​ ,OP​ , and OQ​ are radii of the circle and are therefore congruent. Since ∠SOP and ∠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, ∠SOR and ∠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 andarc 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 144π, the expression x+x+2x+2x is equal to 144π. Thus  x+x+2x+2x = 144π , or 6x = 144π . Dividing both sides of this equation by 6 yields x = 24π. Therefore, the length of arc QR is 2(24π), or 48π.
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.