{"id":"cb_question_bank:8c9abab8-de69-4713-93d3-c72423171dc9","source_id":"cb_question_bank","source_item_id":"8c9abab8-de69-4713-93d3-c72423171dc9","external_id":"84b26634-2244-45e6-8504-ea367a0f2049","ibn":null,"question_id":"c9931030","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 20 upper S upper T equals 48 upper T upper R equals 52 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 20\"><mi>R</mi><mi>S</mi><mo>=</mo><mn>20</mn>\n</math></p>\n<p style=\"text-align: center;\"><math alttext=\"upper S upper T equals 48\"><mi>S</mi><mi>T</mi><mo>=</mo><mn>48</mn>\n</math></p>\n<p style=\"text-align: center;\"><math alttext=\"upper T upper R equals 52\"><mi>T</mi><mi>R</mi><mo>=</mo><mn>52</mn>\n</math></p>\n<p>The side lengths of right triangle <math alttext=\"upper R upper S upper T\"><mi>R</mi><mi>S</mi><mi>T</mi></math> are given. Triangle <math alttext=\"upper R upper S upper T\"><mi>R</mi><mi>S</mi><mi>T</mi></math> is similar to triangle <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>. 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 B is correct. It's given that 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> 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>. It's given that the side lengths of the 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 <math alttext=\"upper R upper S equals 20\"><mrow>\n\t<mrow>\n\t\t<mi>R</mi>\n\t\t<mi>S</mi>\n\t</mrow>\n\t<mo>=</mo>\n\t<mn>20</mn>\n</mrow>\n</math>, <math alttext=\"upper S upper T equals 48\"><mrow>\n\t<mrow>\n\t\t<mi>S</mi>\n\t\t<mi>T</mi>\n\t</mrow>\n\t<mo>=</mo>\n\t<mn>48</mn>\n</mrow>\n</math>, and&nbsp;<math alttext=\"upper T upper R equals 52\"><mi>T</mi><mi>R</mi><mo>=</mo><mn>52</mn></math>. Corresponding angles in similar triangles are equal. It follows that the measure of angle <math alttext=\"upper T\"><mi>T</mi>\n</math> is equal to the measure of angle <math alttext=\"upper W\"><mi>W</mi>\n</math>.&nbsp;The hypotenuse of a right triangle is the longest side. It follows that 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>. The hypotenuse of a right triangle is the side opposite the right angle. Therefore, angle <math alttext=\"upper S\"><mi>S</mi>\n</math> is a right angle. The adjacent side of an acute angle in a right triangle is the side closest to the angle that is not the hypotenuse. It follows that the adjacent side of angle <math alttext=\"upper T\"><mi>T</mi>\n</math> is side <math alttext=\"upper S upper T\"><mrow>\n\t<mi>S</mi>\n\t<mi>T</mi>\n</mrow>\n</math>. The opposite side of an acute angle in a right triangle is the side across from the acute angle. 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\"><mrow>\n\t<mi>R</mi>\n\t<mi>S</mi>\n</mrow>\n</math>. The tangent of an acute angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side. Therefore, <math alttext=\"tangent upper T equals StartFraction upper R upper S Over upper S upper T EndFraction\"><mi>tan</mi><mo>&nbsp;</mo><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=\"20\"><mn>20</mn>\n</math> for <math alttext=\"upper R upper S\"><mrow>\n\t<mi>R</mi>\n\t<mi>S</mi>\n</mrow>\n</math> and <math alttext=\"48\"><mn>48</mn>\n</math> for <math alttext=\"upper S upper T\"><mrow>\n\t<mi>S</mi>\n\t<mi>T</mi>\n</mrow>\n</math> in this equation yields <math alttext=\"tangent upper T equals StartFraction 20 Over 48 EndFraction\"><mi>tan</mi><mo>&nbsp;</mo><mi>T</mi><mo>=</mo><mfrac><mn>20</mn><mn>48</mn></mfrac></math>, or <math alttext=\"tangent upper T equals five twelfths\"><mi>tan</mi><mo>&nbsp;</mo><mi>T</mi><mo>=</mo><mfrac><mn>5</mn><mn>12</mn></mfrac></math>. The tangents of two acute angles with equal measures are equal. Since the measure of angle <math alttext=\"upper T\"><mi>T</mi>\n</math> is equal to the measure of angle <math alttext=\"upper W\"><mi>W</mi>\n</math>, it follows that <math alttext=\"tangent upper T equals tangent upper W\"><mi>tan</mi><mo>&nbsp;</mo><mi>T</mi><mo>=</mo><mi>tan</mi><mo>&nbsp;</mo><mi>W</mi></math>. Substituting <math alttext=\"five twelfths\"><mfrac><mn>5</mn><mn>12</mn></mfrac></math> for&nbsp;<math alttext=\"tangent upper T\"><mi>tan</mi><mo>&nbsp;</mo><mi>T</mi></math> in this equation yields <math alttext=\"five twelfths equals tangent upper W\"><mfrac><mn>5</mn><mn>12</mn></mfrac><mo>=</mo><mi>tan</mi><mo>&nbsp;</mo><mi>W</mi></math>. Therefore, the value of <math alttext=\"tangent upper W\"><mi>tan</mi><mo>&nbsp;</mo><mi>W</mi></math> is <math alttext=\"five twelfths\"><mfrac><mn>5</mn><mn>12</mn></mfrac></math>.</p>\n<p>Choice A is incorrect. This is the value of <math alttext=\"sine upper W\"><mi>sin</mi><mo>&nbsp;</mo><mi>W</mi></math>.</p>\n<p>Choice C is incorrect. This is the value of <math alttext=\"cosine upper W\"><mi>cos</mi><mo>&nbsp;</mo><mi>W</mi></math>.</p>\n<p>Choice D is incorrect. This is the value of <math alttext=\"StartFraction 1 Over tangent upper W EndFraction\"><mfrac><mn>1</mn><mrow><mi>tan</mi><mo>&nbsp;</mo><mi>W</mi></mrow></mfrac></math>.</p>","correct_answer":["B"],"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=\"five thirteenths\"><mfrac>\n\t<mn>5</mn>\n\t<mn>13</mn>\n</mfrac>\n</math></p>","ord":0},{"label":"B","content_html":"<p><math alttext=\"five twelfths\"><mfrac>\n\t<mn>5</mn>\n\t<mn>12</mn>\n</mfrac>\n</math></p>","ord":1},{"label":"C","content_html":"<p><math alttext=\"StartFraction 12 Over 13 EndFraction\"><mfrac>\n\t<mn>12</mn>\n\t<mn>13</mn>\n</mfrac>\n</math></p>","ord":2},{"label":"D","content_html":"<p><math alttext=\"StartFraction 12 Over 5 EndFraction\"><mfrac>\n\t<mn>12</mn>\n\t<mn>5</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}}