{"id":"cb_question_bank:8030f1af-3468-41fa-9db5-43befa97047a","source_id":"cb_question_bank","source_item_id":"8030f1af-3468-41fa-9db5-43befa97047a","external_id":"49178e05-53a1-4bcb-954a-708fc5b770ff","ibn":null,"question_id":"0e709a29","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":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"upper R upper S equals 440 upper S upper T equals 384 upper T upper R equals 584 The side lengths of right triangle upper R upper S upper T are given. Triangle upper R upper S upper T is similar to triangle upper U upper V upper W , where upper S corresponds to upper V and upper T corresponds to upper W . What is the value of tangent upper W ?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: center;\"><math alttext=\"upper R upper S equals 440\"><mi>R</mi><mi>S</mi><mo>=</mo><mn>440</mn>\n</math></p>\n<p style=\"text-align: center;\"><math alttext=\"upper S upper T equals 384\"><mi>S</mi><mi>T</mi><mo>=</mo><mn>384</mn>\n</math></p>\n<p style=\"text-align: center;\"><math alttext=\"upper T upper R equals 584\"><mi>T</mi><mi>R</mi><mo>=</mo><mn>584</mn>\n</math></p>\n<p>The side lengths of right triangle <math alttext=\"upper R upper S upper T\"><mrow>\n\t<mi>R</mi>\n\t<mi>S</mi>\n\t<mi>T</mi>\n</mrow>\n</math> are given. Triangle <math alttext=\"upper R upper S upper T\"><mrow>\n\t<mi>R</mi>\n\t<mi>S</mi>\n\t<mi>T</mi>\n</mrow>\n</math>&nbsp;&nbsp;is similar to triangle <math alttext=\"upper U upper V upper W\"><mrow>\n\t<mi>U</mi>\n\t<mi>V</mi>\n\t<mi>W</mi>\n</mrow>\n</math>, where <math alttext=\"upper S\"><mi>S</mi>\n</math> corresponds to <math alttext=\"upper V\"><mi>V</mi>\n</math> and <math alttext=\"upper T\"><mi>T</mi>\n</math> corresponds to <math alttext=\"upper W\"><mi>W</mi>\n</math>. What is the value of <math alttext=\"tangent upper W\"><mi>tan</mi><mi>W</mi></math>?</p>","rationale_html":"<p style=\"text-align: left;\">Choice D is correct. The hypotenuse of triangle&nbsp;<math alttext=\"upper R upper S upper T\"><mi>R</mi><mi>S</mi><mi>T</mi></math> is the longest side and is across from the right angle. The longest side length given is <math alttext=\"584\"><mn>584</mn>\n</math>, which is the length of side&nbsp;<math alttext=\"upper T upper R\"><mi>T</mi><mi>R</mi></math>. Therefore, the hypotenuse of triangle&nbsp;<math alttext=\"upper R upper S upper T\"><mi>R</mi><mi>S</mi><mi>T</mi></math> is side <math alttext=\"upper T upper R\"><mi>T</mi><mi>R</mi></math>, so the right angle is angle <math alttext=\"upper S\"><mi>S</mi>\n</math>. The tangent of an acute angle in a right triangle is the ratio of the length of the opposite side, which is the side across from the angle, to the length of the adjacent side, which is the side closest to the angle that is not the hypotenuse. It follows that the opposite side of angle <math alttext=\"upper T\"><mi>T</mi>\n</math> is side <math alttext=\"upper R upper S\"><mi>R</mi><mi>S</mi></math> and the adjacent side of angle <math alttext=\"upper T\"><mi>T</mi>\n</math> is side <math alttext=\"upper S upper T\"><mi>S</mi><mi>T</mi></math>. Therefore, <math alttext=\"tangent upper T equals StartFraction upper R upper S Over upper S upper T EndFraction\"><mi>tan</mi><mi>T</mi><mo>=</mo><mfrac><mrow><mi>R</mi><mi>S</mi></mrow><mrow><mi>S</mi><mi>T</mi></mrow></mfrac></math>. Substituting <math alttext=\"440\"><mn>440</mn>\n</math> for&nbsp;<math alttext=\"upper R upper S\"><mi>R</mi><mi>S</mi></math> and <math alttext=\"384\"><mn>384</mn>\n</math> for&nbsp;<math alttext=\"upper S upper T\"><mi>S</mi><mi>T</mi></math> in this equation yields <math alttext=\"tangent upper T equals StartFraction 440 Over 384 EndFraction\"><mi>tan</mi><mi>T</mi><mo>=</mo><mfrac><mn>440</mn><mn>384</mn></mfrac></math>. This is equivalent to&nbsp;<math alttext=\"tangent upper T equals StartFraction 55 Over 48 EndFraction\"><mi>tan</mi><mi>T</mi><mo>=</mo><mfrac><mn>55</mn><mn>48</mn></mfrac></math>. It’s given that triangle <math alttext=\"upper R upper S upper T\"><mi>R</mi><mi>S</mi><mi>T</mi></math> is similar to triangle&nbsp;<math alttext=\"upper U upper V upper W\"><mi>U</mi><mi>V</mi><mi>W</mi></math>, where <math alttext=\"upper S\"><mi>S</mi>\n</math> corresponds to <math alttext=\"upper V\"><mi>V</mi>\n</math> and <math alttext=\"upper T\"><mi>T</mi>\n</math> corresponds to <math alttext=\"upper W\"><mi>W</mi>\n</math>. It follows that <math alttext=\"upper R\"><mi>R</mi>\n</math> corresponds to <math alttext=\"upper U\"><mi>U</mi>\n</math>. Therefore, the hypotenuse of triangle <math alttext=\"upper U upper V upper W\"><mi>U</mi><mi>V</mi><mi>W</mi></math> is side <math alttext=\"upper W upper U\"><mi>W</mi><mi>U</mi></math>, which means <math alttext=\"tangent upper W equals StartFraction upper U upper V Over upper V upper W EndFraction\"><mi>tan</mi><mi>W</mi><mo>=</mo><mfrac><mrow><mi>U</mi><mi>V</mi></mrow><mrow><mi>V</mi><mi>W</mi></mrow></mfrac></math>. Since the lengths of corresponding sides of similar triangles are proportional, <math alttext=\"StartFraction upper R upper S Over upper S upper T EndFraction equals StartFraction upper U upper V Over upper V upper W EndFraction\"><mfrac><mrow><mi>R</mi><mi>S</mi></mrow><mrow><mi>S</mi><mi>T</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>U</mi><mi>V</mi></mrow><mrow><mi>V</mi><mi>W</mi></mrow></mfrac></math>. Therefore, <math alttext=\"tangent upper W equals StartFraction upper U upper V Over upper V upper W EndFraction\"><mi>tan</mi><mi>W</mi><mo>=</mo><mfrac><mrow><mi>U</mi><mi>V</mi></mrow><mrow><mi>V</mi><mi>W</mi></mrow></mfrac></math> is equivalent to <math alttext=\"tangent upper W equals StartFraction upper R upper S Over upper S upper T EndFraction\"><mi>tan</mi><mi>W</mi><mo>=</mo><mfrac><mrow><mi>R</mi><mi>S</mi></mrow><mrow><mi>S</mi><mi>T</mi></mrow></mfrac></math>, or <math alttext=\"tangent upper W equals tangent upper T\"><mi>tan</mi><mi>W</mi><mo>=</mo><mi>tan</mi><mi>T</mi></math>. Thus, <math alttext=\"tangent upper W equals StartFraction 55 Over 48 EndFraction\"><mi>tan</mi><mi>W</mi><mo>=</mo><mfrac><mn>55</mn><mn>48</mn></mfrac></math>.</p>\n<p style=\"text-align: left;\">Choice A is incorrect. This is the value of <math alttext=\"cosine upper W\"><mi>cos</mi><mi>W</mi></math>, not <math alttext=\"tangent upper W\"><mi>tan</mi><mi>W</mi></math>.</p>\n<p style=\"text-align: left;\">Choice B is incorrect. This is the value of <math alttext=\"sine upper W\"><mi>sin</mi><mi>W</mi></math>, not <math alttext=\"tangent upper W\"><mi>tan</mi><mi>W</mi></math>.</p>\n<p style=\"text-align: left;\">Choice C is incorrect. This is the value of <math alttext=\"StartFraction 1 Over tangent upper W EndFraction\"><mfrac><mn>1</mn><mrow><mi>tan</mi><mi>W</mi></mrow></mfrac></math>, not <math alttext=\"tangent upper W\"><mi>tan</mi><mi>W</mi></math>.</p>","correct_answer":["D"],"has_media":false,"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":"<p><math alttext=\"StartFraction 48 Over 73 EndFraction\"><mfrac>\n\t<mn>48</mn>\n\t<mn>73</mn>\n</mfrac>\n</math></p>","ord":0},{"label":"B","content_html":"<p><math alttext=\"StartFraction 55 Over 73 EndFraction\"><mfrac>\n\t<mn>55</mn>\n\t<mn>73</mn>\n</mfrac>\n</math></p>","ord":1},{"label":"C","content_html":"<p><math alttext=\"StartFraction 48 Over 55 EndFraction\"><mfrac>\n\t<mn>48</mn>\n\t<mn>55</mn>\n</mfrac>\n</math></p>","ord":2},{"label":"D","content_html":"<p><math alttext=\"StartFraction 55 Over 48 EndFraction\"><mfrac>\n\t<mn>55</mn>\n\t<mn>48</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}}