The circle has center O, circumference 144π,and diameters PR and QS. The length of arc PS is twice the length of arc PQ. What is the length of arc QR ?

Explanation

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 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 144π , the expression x + x +2x +2x isequal to 144π . Thus xx x x ++ + = 2 2 144π , or 6 144 x = π. Dividing both sides ofthis equation by 6 yields x =24π . Therefore, the length of arc QR is 2 24 π , or 48.