{"id":"cb_question_bank:2e3f10e4-a193-467a-bbef-16e6fd5baabc","source_id":"cb_question_bank","source_item_id":"2e3f10e4-a193-467a-bbef-16e6fd5baabc","external_id":null,"ibn":"029664-DC","question_id":"bd87bc09","program":"SAT","module":"math","domain_code":"S","domain":"Geometry and Trigonometry","skill_code":"S.C.","skill":"Right triangles and trigonometry","difficulty":"H","score_band":6,"answer_type":"spr","answer_source":"rationale","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"Triangle A, B C above is a right triangle, and the sine of B, equals 5 over 13 . What is the length of side B C ?","stimulus":"The figure presents right triangle A, B C. Side A, C is horizontal, and vertex B is directly above vertex C. Angle C is a right angle. The length of side A, B is 26.","stimulus_html":"<div class=\"stimulus_reference \">\n        <div class=\"standalone_image \">\n          <img src=\"/api/assets/55e0715b9341977adb8761b19c5c1d4e23105bfecc3836e849aac704a24b736d\" alt=\"The figure presents right triangle A, B C. Side A, C is horizontal, and vertex B is directly above vertex C. Angle C is a right angle. The length of side A, B is 26.\" width=\"58\" height=\"115\"></div>\n      </div>\n","stem_html":"<p class=\"stem_paragraph \">Triangle <span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/ac7cf5fed128f7d6811702ead19d3daa22268a1a0bc26dcab050c84f3942a94f\" alt=\"A, B C\"></span></span> above is a right triangle, and <span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/7ee5df65d52341e9c5654da90fad0b21f769cea7b61bd8cc6b8ef762252a9ee2\" alt=\"the sine of B, equals 5 over 13\"></span></span>. What is the length of side <span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/28ddeb9057580c7e30bf35d0a356276556d1018a2bc2bc33e2c76d746b8a8cfa\" alt=\"B C\"></span></span>?</p>\n","rationale_html":"<p>The correct answer is 24. The sine of an acute angle in a right triangle is equal to the ratio of the length of the side opposite the angle to the length of the hypotenuse. In the triangle shown, the sine of angle <span class=\"italic\">B</span>, or <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/bb5da72960301bec754c9bc11cfd461e31c38f7e4e9eaced62dfd7237da96770\" alt=\"sine of B\"></span>, is equal to the ratio of the length of side <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/838c37eb26f8e7bea94de28d37cede03fda6bc1cb7121288f537103ca3516078\" alt=\"A, C\"></span> to the length of side <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/bc38232964225d3fa314e7d0470d11422beea36b6598d546923dc085e690921c\" alt=\"A, B\"></span><span class=\"italic\">.</span> It’s given that the length of side <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/bc38232964225d3fa314e7d0470d11422beea36b6598d546923dc085e690921c\" alt=\"A, B\"></span> is 26 and that <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/7ee5df65d52341e9c5654da90fad0b21f769cea7b61bd8cc6b8ef762252a9ee2\" alt=\"sine of B equals, 5 over 13\"></span>. Therefore, <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/6305ffe91fb9ca1ca57f6139d8f4f1eff8cef0c5867776dc54b7f7cd6f53b67f\" alt=\"the fraction 5 over 13, equals, the fraction A, B over 26\"></span>. Multiplying both sides of this equation by 26 yields <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/0b6b158728a87744d0443c609ec9602227e8eddc019ebcb73810f38cb0841467\" alt=\"A, C equals 10\"></span>.</p><p>By the Pythagorean Theorem, the relationship between the lengths of the sides of triangle <span class=\"italic\">ABC</span> is as follows: <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/5ef2931b859a1d2ab0ae4e71256457679d3776c4178f3347c9d5b062f590de8d\" alt=\"26 squared equals, 10 squared, plus B C squared\"></span>, or <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/e91cd90b65ad8f2d90e69aa77e950ef1b4268da7f7871ba662ad4bd38776dda8\" alt=\"676 equals, 100 plus B C squared\"></span>. Subtracting 100 from both sides of <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/e91cd90b65ad8f2d90e69aa77e950ef1b4268da7f7871ba662ad4bd38776dda8\" alt=\"676 equals, 100 plus B C squared\"></span> yields <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/7d143e04ae4bef1fb36e11b3bd5c16e62397ca95cb6eccb3aac017dd984915e3\" alt=\"576 equals, B C squared\"></span>. Taking the square root of both sides of <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/7d143e04ae4bef1fb36e11b3bd5c16e62397ca95cb6eccb3aac017dd984915e3\" alt=\"576 equals, B C squared\"></span> yields <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/97fc082c6901798cf1de7b00da3a40fc195116036fa462387fc729ec3e1c62e9\" alt=\"24 equals B C\"></span>.</p><p></p>\n","correct_answer":["24"],"has_media":true,"has_mathml":false,"has_table":false,"source_created_at":"2023-08-02T20:25:59.626000+00:00","source_updated_at":"2023-08-02T20:25:59.626000+00:00","options":[],"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}}