{"id":"cb_question_bank:b99464f1-d5c9-41d9-87f7-b453fffd29f2","source_id":"cb_question_bank","source_item_id":"b99464f1-d5c9-41d9-87f7-b453fffd29f2","external_id":"28b7a76a-9194-4446-b1c9-e00fb73bc3c8","ibn":null,"question_id":"0815a5af","program":"SAT","module":"math","domain_code":"S","domain":"Geometry and Trigonometry","skill_code":"S.D.","skill":"Circles","difficulty":"M","score_band":4,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"The center of the circle is point upper O. Points upper S, upper R, upper Q, and upper P are on the circle. Line segment upper P upper R is a diameter of the circle. Line segment upper Q upper S is a diameter of the circle. Diameters upper P upper R and upper Q upper S intersect at point upper O. A note indicates the figure is not drawn to scale. The circle shown has center upper O , circumference 144 pi , and diameters line segment upper P upper R and line segment upper Q upper S . The length of arc upper P upper S is twice the length of arc upper P upper Q . What is the length of arc upper Q upper R ?","stimulus":"","stimulus_html":null,"stem_html":"

\n

The circle shown has center O\n, circumference 144π, and diameters PR¯ and QS¯. The length of arc \n\tP\n\tS\n\n is twice the length of arc \n\tP\n\tQ\n\n. What is the length of arc \n\tQ\n\tR\n\n?

","rationale_html":"

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\n 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\n. 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 is equal to 144π. Thus x+x+2x+2x=144π, or 6x=144π. Dividing both sides of this equation by 6\n yields x=24π. Therefore, the length of arc QR is 224π, or 48π.

\n

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

\n

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

\n

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

","correct_answer":["B"],"has_media":true,"has_mathml":true,"has_table":false,"source_created_at":"2023-08-02T20:25:59.840000+00:00","source_updated_at":"2023-08-02T20:25:59.840000+00:00","options":[{"label":"A","content_html":"

24π

","ord":0},{"label":"B","content_html":"

48π

","ord":1},{"label":"C","content_html":"

72π

","ord":2},{"label":"D","content_html":"

96π

","ord":3}],"attribution":{"source_id":"cb_question_bank","source_name":"College Board digital SAT question bank (community dump)","rights_holder":"College Board","canonical_url":"https://satsuitequestionbank.collegeboard.org/","retrieved_from":"https://raw.githubusercontent.com/mdn522/sat-question-bank/main/data/cb-digital-questions.json","attribution":"SAT practice questions © College Board, from the official SAT Suite Question Bank. Retrieved via the community dump at github.com/mdn522/sat-question-bank.","disclaimer":"SAT® and PSAT/NMSQT® are trademarks registered by the College Board, which is not affiliated with, does not sponsor, and does not endorse this project.","license":"College Board copyright; source repository carries no licence. Local use only.","redistributable":false,"item_count":2017}}