The center of the circle is point upper O. Points upper S, upper R, upper Q, and upper P are

Question

The circle shown has center $O$ , circumference $144 pi$ , and diameters $\bar{P R}$ and $\bar{Q S}$ . The length of arc $P S$ is twice the length of arc $P Q$ . What is the length of arc $Q R$ ?

Accepted Answer

Choice B is correct. Since P R /mo> and Q S /mo> are diameters of the circle shown, O S /mo> , O R /mo> , O P /mo> , and O Q /mo> are radii of the circle and are therefore congruent. Since S O P and R O Q are vertical angles, they are congruent. Therefore, arc P S and arc Q R are formed by congruent radii and have the same angle measure, so they are congruent arcs. Similarly, S O R and P O Q are vertical angles, so they are congruent. Therefore, arc S R and arc P Q are formed by congruent radii and have the same angle measure, so they are congruent arcs. Let x represent the length of arc S R . Since arc S R and arc P Q are congruent arcs, the length of arc P Q can also be represented by x . It&rsquo;s given that the length of arc P S is twice the length of arc P Q . Therefore, the length of arc P S can be represented by the expression 2 x . Since arc P S and arc Q R are congruent arcs, the length of arc Q R can also be represented by 2 x . This gives the expression x + x + 2 x + 2 x . Since it's given that the circumference is 144 , the expression x + x + 2 x + 2 x is equal to 144 . Thus x + x + 2 x + 2 x = 144 , or 6 x = 144 . Dividing both sides of this equation by 6 yields x = 24 . Therefore, the length of arc Q R is 2 24 , or 48 . Choice A is incorrect. This is the length of arc P Q , not arc Q R . Choice C is incorrect and may result from conceptual or calculation errors. Choice D is incorrect and may result from conceptual or calculation errors.

Question 28b7a7