{"id":"cb_question_bank:20928cc3-f440-48bf-90d4-e60e8f4560f1","source_id":"cb_question_bank","source_item_id":"20928cc3-f440-48bf-90d4-e60e8f4560f1","external_id":"6c28e838-bf2a-4be6-ba95-64b7eb7c520d","ibn":null,"question_id":"a0369739","program":"SAT","module":"math","domain_code":"S","domain":"Geometry and Trigonometry","skill_code":"S.B.","skill":"Lines, angles, and triangles","difficulty":"H","score_band":7,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"In triangle upper A upper B upper C , the measure of angle upper B is 90 degree and line segment upper B upper D is an altitude of the triangle. The length of line segment upper A upper B is 15 and the length of line segment upper A upper C is 23 greater than the length of line segment upper A upper B . What is the value of StartFraction upper B upper C Over upper B upper D EndFraction ?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: left;\">In triangle <math alttext=\"upper A upper B upper C\"><mrow>\n\t<mi>A</mi>\n\t<mi>B</mi>\n\t<mi>C</mi>\n</mrow>\n</math>, the measure of angle <math alttext=\"upper B\"><mi>B</mi>\n</math> is <math alttext=\"90 degree\"><mn>90</mn><mo>°</mo></math> and <math alttext=\"line segment upper B upper D\"><menclose notation=\"top\"><mi>B</mi><mi>D</mi></menclose></math> is an altitude of the triangle. The length of <math alttext=\"line segment upper A upper B\"><menclose notation=\"top\"><mi>A</mi><mi>B</mi></menclose></math> is <math alttext=\"15\"><mn>15</mn>\n</math> and the length of <math alttext=\"line segment upper A upper C\"><menclose notation=\"top\"><mi>A</mi><mi>C</mi></menclose></math> is <math alttext=\"23\"><mn>23</mn>\n</math> greater than the length of <math alttext=\"line segment upper A upper B\"><menclose notation=\"top\"><mi>A</mi><mi>B</mi></menclose></math>. What is the value of <math alttext=\"StartFraction upper B upper C Over upper B upper D EndFraction\"><mfrac><mrow><mi>B</mi><mi>C</mi></mrow><mrow><mi>B</mi><mi>D</mi></mrow></mfrac></math>?</p>","rationale_html":"<p style=\"text-align: left;\">Choice D is correct. It's given that in triangle&nbsp;<math alttext=\"upper A upper B upper C\"><mi>A</mi><mi>B</mi><mi>C</mi></math>, the measure of angle <math alttext=\"upper B\"><mi>B</mi>\n</math> is&nbsp;<math alttext=\"90 degree\"><mn>90</mn><mo>°</mo></math> and&nbsp; <math alttext=\"line segment upper B upper D\"><menclose notation=\"top\"><mi>B</mi><mi>D</mi></menclose></math> is an altitude of the triangle. Therefore, the measure of angle <math alttext=\"upper B upper D upper C\"><mi>B</mi><mi>D</mi><mi>C</mi></math> is <math alttext=\"90 degree\"><mn>90</mn><mo>°</mo></math>. It follows that angle <math alttext=\"upper B\"><mi>B</mi>\n</math> is congruent to angle <math alttext=\"upper D\"><mi>D</mi>\n</math> and angle <math alttext=\"upper C\"><mi>C</mi>\n</math> is congruent to angle <math alttext=\"upper C\"><mi>C</mi>\n</math>. By the angle-angle similarity postulate, triangle&nbsp;<math alttext=\"upper A upper B upper C\"><mi>A</mi><mi>B</mi><mi>C</mi></math> is similar to triangle <math alttext=\"upper B upper D upper C\"><mi>B</mi><mi>D</mi><mi>C</mi></math>. Since triangles <math alttext=\"upper A upper B upper C\"><mi>A</mi><mi>B</mi><mi>C</mi></math> and&nbsp;<math alttext=\"upper B upper D upper C\"><mi>B</mi><mi>D</mi><mi>C</mi></math> are similar, it follows that&nbsp;<math alttext=\"StartFraction upper A upper C Over upper A upper B EndFraction equals StartFraction upper B upper C Over upper B upper D EndFraction\"><mfrac><mrow><mi>A</mi><mi>C</mi></mrow><mrow><mi>A</mi><mi>B</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>B</mi><mi>C</mi></mrow><mrow><mi>B</mi><mi>D</mi></mrow></mfrac></math>. It's also given that the length of&nbsp;<math alttext=\"line segment upper A upper B\"><mover><mrow><mi>A</mi><mi>B</mi></mrow><mo>¯</mo></mover></math> is <math alttext=\"15\"><mn>15</mn>\n</math> and the length of&nbsp;<math alttext=\"line segment upper A upper C\"><mover><mrow><mi>A</mi><mi>C</mi></mrow><mo>¯</mo></mover></math> is <math alttext=\"23\"><mn>23</mn>\n</math> greater than the length of <math alttext=\"line segment upper A upper B\"><mover><mrow><mi>A</mi><mi>B</mi></mrow><mo>¯</mo></mover></math>. Therefore, the length of&nbsp;<math alttext=\"line segment upper A upper C\"><mover><mrow><mi>A</mi><mi>C</mi></mrow><mo>¯</mo></mover></math> is <math alttext=\"15 plus 23\"><mn>15</mn><mo>+</mo><mn>23</mn></math>, or <math alttext=\"38\"><mn>38</mn>\n</math>. Substituting <math alttext=\"15\"><mn>15</mn>\n</math> for&nbsp;<math alttext=\"upper A upper B\"><mi>A</mi><mi>B</mi></math> and <math alttext=\"38\"><mn>38</mn>\n</math> for&nbsp;<math alttext=\"upper A upper C\"><mi>A</mi><mi>C</mi></math> in the equation <math alttext=\"StartFraction upper A upper C Over upper A upper B EndFraction equals StartFraction upper B upper C Over upper B upper D EndFraction\"><mfrac><mrow><mi>A</mi><mi>C</mi></mrow><mrow><mi>A</mi><mi>B</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>B</mi><mi>C</mi></mrow><mrow><mi>B</mi><mi>D</mi></mrow></mfrac></math> yields <math alttext=\"StartFraction 38 Over 15 EndFraction equals StartFraction upper B upper C Over upper B upper D EndFraction\"><mfrac><mn>38</mn><mn>15</mn></mfrac><mo>=</mo><mfrac><mrow><mi>B</mi><mi>C</mi></mrow><mrow><mi>B</mi><mi>D</mi></mrow></mfrac></math>. Therefore, the value of&nbsp;<math alttext=\"StartFraction upper B upper C Over upper B upper D EndFraction\"><mfrac><mrow><mi>B</mi><mi>C</mi></mrow><mrow><mi>B</mi><mi>D</mi></mrow></mfrac></math> is&nbsp;<math alttext=\"StartFraction 38 Over 15 EndFraction\"><mfrac><mn>38</mn><mn>15</mn></mfrac></math>.</p>\n<p>Choice A is incorrect. This is the value of <math alttext=\"StartFraction upper B upper D Over upper B upper C EndFraction\"><mfrac><mrow><mi>B</mi><mi>D</mi></mrow><mrow><mi>B</mi><mi>C</mi></mrow></mfrac></math>.</p>\n<p>Choice B is incorrect and may result from conceptual or calculation errors.</p>\n<p>Choice C is incorrect and may result from conceptual or calculation errors.<br>&nbsp;</p>","correct_answer":["D"],"has_media":false,"has_mathml":true,"has_table":false,"source_created_at":"2023-08-02T20:25:59.837000+00:00","source_updated_at":"2023-08-02T20:25:59.837000+00:00","options":[{"label":"A","content_html":"<p><math alttext=\"StartFraction 15 Over 38 EndFraction\"><mfrac>\n\t<mn>15</mn>\n\t<mn>38</mn>\n</mfrac>\n</math></p>","ord":0},{"label":"B","content_html":"<p><math alttext=\"StartFraction 15 Over 23 EndFraction\"><mfrac>\n\t<mn>15</mn>\n\t<mn>23</mn>\n</mfrac>\n</math></p>","ord":1},{"label":"C","content_html":"<p><math alttext=\"StartFraction 23 Over 15 EndFraction\"><mfrac>\n\t<mn>23</mn>\n\t<mn>15</mn>\n</mfrac>\n</math></p>","ord":2},{"label":"D","content_html":"<p><math alttext=\"StartFraction 38 Over 15 EndFraction\"><mfrac>\n\t<mn>38</mn>\n\t<mn>15</mn>\n</mfrac>\n</math></p>","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}}