{"id":"cb_question_bank:fc5135c2-0997-4826-b1f6-8c0d6f53e223","source_id":"cb_question_bank","source_item_id":"fc5135c2-0997-4826-b1f6-8c0d6f53e223","external_id":null,"ibn":"09173-DC","question_id":"25da87f8","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":"A triangle with angle measures 30°, 60°, and 90° has a perimeter of 18 plus 6, times, the square root of 3 . What is the length of the longest side of the triangle?","stimulus":"","stimulus_html":null,"stem_html":"<p class=\"stem_paragraph \">A triangle with angle measures 30°, 60°, and 90° has a perimeter of <span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/8b1600c9998b0cfa9496be652307de61c7b5928394bd3337a1bdc80f5f181adc\" alt=\"18 plus 6, times, the square root of 3\"></span></span>. What is the length of the longest side of the triangle?</p>\n","rationale_html":"<p>The correct answer is 12. It is given that the triangle has angle measures of 30°, 60°, and 90°, and so the triangle is a special right triangle. The side measures of this type of special triangle are in the ratio <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/2bb0da7923068d423781c6c04a32900ab7725391db7e5cc0df504faaee3be681\" alt=\"2 to 1 to the square root of 3\"></span>. If <span class=\"italic\">x</span> is the measure of the shortest leg, then the measure of the other leg is <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/5264b8fa4acffbdb1bf3ec735ac3540a55c42e75f05977258f8a96442c882477\" alt=\"the square root of 3 end root x\"></span> and the measure of the hypotenuse is 2<span class=\"italic\">x</span>. The perimeter of the triangle is given to be <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/8b1600c9998b0cfa9496be652307de61c7b5928394bd3337a1bdc80f5f181adc\" alt=\"18, plus 6 times, the square root of 3, end root\"></span>, and so the equation for the perimeter can be written as <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/f15e3e334cc1c523b32b3d7a268496f5afd0800167754fa14f08f0aadc85cdeb\" alt=\"2 x, plus x, plus the square root of 3 end root x, equals 18, plus 6 times, the square root of 3\"></span>. Combining like terms and factoring out a common factor of <span class=\"italic\">x</span> on the left-hand side of the equation gives <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/1dd8830de2856ba96886357a54a634feba6fafae0fa3b46f3e00cb36acf57386\" alt=\"open parenthesis, 3 plus the square root of 3, close parenthesis, times x, equals 18, plus 6 times the square root of 3\"></span>. Rewriting the right-hand side of the equation by factoring out 6 gives <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/bdbc3b2d9e5b55a8e5835a05a0a66ad8d43d9f45ea99711ef048b2b5ca8bfb70\" alt=\"open parenthesis, 3 plus the square root of 3, close parenthesis, times x, equals 6 times, open parenthesis, 3 plus the square root of 3, close parenthesis\"></span>. Dividing both sides of the equation by the common factor <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/8b9c35655396aa6ce107dc6c1a1a87ea6be3a1ae74300ce53d6cee8a4d7ab5d7\" alt=\"open parenthesis, 3 plus the square root of 3, close parenthesis\"></span> gives <span class=\"italic\">x</span> = 6. The longest side of the right triangle, the hypotenuse, has a length of 2<span class=\"italic\">x</span>, or 2(6), which is 12.</p>\n","correct_answer":["12"],"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":[],"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}}