{"id":"cb_question_bank:2c4a70ab-20ce-4e0e-add7-2bf06bc1263e","source_id":"cb_question_bank","source_item_id":"2c4a70ab-20ce-4e0e-add7-2bf06bc1263e","external_id":null,"ibn":"08415-DC","question_id":"c8345903","program":"SAT","module":"math","domain_code":"S","domain":"Geometry and Trigonometry","skill_code":"S.D.","skill":"Circles","difficulty":"H","score_band":6,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"The circle above has center O , the length of arc A, D C is 5 pi , and x equals 100 . What is the length of arc A, B C ?","stimulus":"The figure presents a circle with center O. There are four points on the circle. Going clockwise around the circle, the four points are A, B, C, and D. Points A and C divide the circle into two arcs, arc A, D C and arc A B C. The length of arc A D C is less than the length of arc A, B C. Radius O A and radius O C are drawn. Angle A O C, the central angle corresponding to arc A, D C, measures x degrees","stimulus_html":"<div class=\"stimulus_reference \">\n        <div class=\"standalone_image \">\n          <img src=\"/api/assets/4fb63557b86e528ea8e6a0236cce7c71fcbcb53052661ac6e7d54ac5548ca891\" alt=\"The figure presents a circle with center O. There are four points on the circle. Going clockwise around the circle, the four points are A, B, C, and D.  Points A and C divide the circle into two arcs, arc A, D C and arc A B C. The length of arc A D C is less than the length of arc A, B C. Radius O A and radius O C are drawn. Angle A O C, the central angle corresponding to arc A, D C, measures x degrees\" width=\"85\" height=\"76\"></div>\n      </div>\n","stem_html":"<p class=\"stem_paragraph \">The circle above has center <span class=\"italic\">O</span>, the length of arc <span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/351482bb52386df11f4b32b97973eb30e0accdc9bd69e042538925adb789aeb5\" alt=\"A, D C\"></span></span> is <span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/87cb24cd8669cb58317bac540ea6738f25f533ab7fc441c5f280c2c32ab4a336\" alt=\"5 pi\"></span></span>, and <span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/e89a11a5938631839440ee70e628fad11515b238437b8f95c02c75b11dca3ea5\" alt=\"x equals 100\"></span></span>. What is the length of arc <span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/126cf3f10285af72b6840dc63f9796980a6466bd150aab551990b5c0a5b1fd6e\" alt=\"A, B C\"></span></span> ?</p>\n","rationale_html":"<p>Choice B is correct. The ratio of the lengths of two arcs of a circle is equal to the ratio of the measures of the central angles that subtend the arcs. It’s given that arc <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/b39b5c9e1758806ba86774452f96aa7ef493de61a609e487bf0a9c713e0b7e9f\" alt=\"A D C\"></span> is subtended by a central angle with measure 100°. Since the sum of the measures of the angles about a point is 360°, it follows that arc <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/1659d497bfd5ea47eecc8ce276900ab8a86c7a767472bcb18d3019f5287c6a2a\" alt=\"A B C\"></span> is subtended by a central angle with measure <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/455d5dbfd8ca3e32745263309c776d287962ee57d5b5b688c5244607edd96f51\" alt=\"360 degrees minus 100 degrees, equals 260 degrees\"></span>. If <span class=\"italic\">s</span> is the length of arc <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/1659d497bfd5ea47eecc8ce276900ab8a86c7a767472bcb18d3019f5287c6a2a\" alt=\"A B C\"></span>, then <span class=\"italic\">s</span> must satisfy the ratio <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/534277656bbce769a8995fc69dfe75dcc22c783aa698dd9289ff638adb96cff0\" alt=\"the fraction s over 5 pi, end fraction equals, the fraction 260 over 100\"></span>. Reducing the fraction <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/f11dface94a53851febe33f6eecbe9e777a34c43cdcd843f255995de37c8569d\" alt=\"260 over 100\"></span> to its simplest form gives <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/7cf604dcda36c53cde65396092d3b943e321f4bc945f4980e03f4efd3d7115a6\" alt=\"the fraction 13 over 5\"></span>. Therefore, <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/9707553265018bf0f50502e989f87a011a8677edcb19b942a3f3fe7b9dd9b687\" alt=\"the fraction s over 5 pi, end fraction, equals, the fraction 13 over 5\"></span>. Multiplying both sides of <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/9707553265018bf0f50502e989f87a011a8677edcb19b942a3f3fe7b9dd9b687\" alt=\"the fraction s over 5 pi, end fraction, equals, the fraction 13 over 5\"></span> by <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/e948b9badb1c98f562ef4a6ff4b29c6732928b862dc8a1ae59e819991a9c37c6\" alt=\"5 pi\"></span> yields <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/11cdee0b13f79250b58a42f08212cffa40df7e24aa0291c04a713cbdef6340ec\" alt=\"s equals 13 pi\"></span>.</p><p>Choice A is incorrect. This is the length of an arc consisting of exactly half of the circle, but arc <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/1659d497bfd5ea47eecc8ce276900ab8a86c7a767472bcb18d3019f5287c6a2a\" alt=\"A B C\"></span> is greater than half of the circle. Choice C is incorrect. This is the total circumference of the circle. Choice D is incorrect. This is half the length of arc <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/1659d497bfd5ea47eecc8ce276900ab8a86c7a767472bcb18d3019f5287c6a2a\" alt=\"A B C\"></span>, not its full length.</p><p></p>\n","correct_answer":["B"],"has_media":true,"has_mathml":false,"has_table":false,"source_created_at":"2023-08-02T20:25:59.636000+00:00","source_updated_at":"2023-08-02T20:25:59.636000+00:00","options":[{"label":"A","content_html":"<p class=\"choice_paragraph \">\n            <span class=\"math_expression \">\n              <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/5c33aa160428129bfb1c2155ad4ffaf58f85ea31e3a652bea488965bd18011de\" alt=\"9 pi\"></span></span>\n          </p>\n","ord":0},{"label":"B","content_html":"<p class=\"choice_paragraph \">\n            <span class=\"math_expression \">\n              <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/60692639d9bd2c170850f5e70bdbeb7b9c21f2f8f08c0464fc5f03aae74fe826\" alt=\"13 pi\"></span></span>\n          </p>\n","ord":1},{"label":"C","content_html":"<p class=\"choice_paragraph \">\n            <span class=\"math_expression \">\n              <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/a48395ce5114d452bea7d616fa7380b91022ded1e53cd47039d46450bece19b4\" alt=\"18 pi\"></span></span>\n          </p>\n","ord":2},{"label":"D","content_html":"<p class=\"choice_paragraph \">\n            <span class=\"math_expression \">\n              <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/6d5777e10e74e0e3fe77f0e44f9e9d89205feab5f09099e0dbb116867b622db8\" alt=\"13 halves pi\"></span></span>\n          </p>\n","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}}