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Note: Figure not drawn to scale. 144π The circle shown has center O, circumference, and diameters PR and QS. The length of arc PS is twice the length of arc PQ. What is the length of arc QR? 24π
Correct answer: B (parsed from the explanation — this source omits an answer key for this item)
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 144r + + + 144r the circumference is , the expression x x 2x 2x is equal to . Thus + +2 +2 =144r 6 =144r x x x x , or x . Dividing both sides of this equation by 6 =24r 2 24r 48r yields x . Therefore, the length of arc QR is ^ h, or . 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.
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