{"id":"cb_question_bank:c305f449-2384-4a03-9630-d376958ac357","source_id":"cb_question_bank","source_item_id":"c305f449-2384-4a03-9630-d376958ac357","external_id":"2ec14ac8-bf5b-46cc-8993-b907b006b31c","ibn":null,"question_id":"c6dff223","program":"SAT","module":"math","domain_code":"S","domain":"Geometry and Trigonometry","skill_code":"S.C.","skill":"Right triangles and trigonometry","difficulty":"H","score_band":7,"answer_type":"spr","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"Triangle upper A upper B upper C is similar to triangle upper D upper E upper F , where angle upper A corresponds to angle upper D and angles upper C and upper F are right angles. The length of line segment upper A upper B is 2.9 times the length of line segment upper D upper E . If tangent upper A equals StartFraction 21 Over 20 EndFraction , what is the value of sine upper D ?","stimulus":"","stimulus_html":null,"stem_html":"<p>Triangle&nbsp;<math alttext=\"upper A upper B upper C\"><mi>A</mi><mi>B</mi><mi>C</mi></math> is similar to triangle&nbsp;<math alttext=\"upper D upper E upper F\"><mi>D</mi><mi>E</mi><mi>F</mi></math>, where angle <math alttext=\"upper A\"><mi>A</mi>\n</math> corresponds to angle <math alttext=\"upper D\"><mi>D</mi>\n</math> and angles <math alttext=\"upper C\"><mi>C</mi>\n</math> and <math alttext=\"upper F\"><mi>F</mi>\n</math> are right angles. 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=\"2.9\"><mn>2.9</mn>\n</math> times the length of&nbsp;<math alttext=\"line segment upper D upper E\"><mover><mrow><mi>D</mi><mi>E</mi></mrow><mo>¯</mo></mover></math>. If&nbsp;<math alttext=\"tangent upper A equals StartFraction 21 Over 20 EndFraction\"><mi>tan</mi><mi>A</mi><mo>=</mo><mfrac><mrow><mn>21</mn></mrow><mrow><mn>20</mn></mrow></mfrac></math>, what is the value of&nbsp;<math alttext=\"sine upper D\"><mi>sin</mi><mi>D</mi></math>?</p>","rationale_html":"<p style=\"text-align: left;\">The correct answer is <math alttext=\"StartFraction 21 Over 29 EndFraction\"><mfrac>\n\t<mn>21</mn>\n\t<mn>29</mn>\n</mfrac>\n</math>. It's given that 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> is similar to triangle <math alttext=\"upper D upper E upper F\"><mrow>\n\t<mi>D</mi>\n\t<mi>E</mi>\n\t<mi>F</mi>\n</mrow>\n</math>, where angle <math alttext=\"upper A\"><mi>A</mi>\n</math> corresponds to angle <math alttext=\"upper D\"><mi>D</mi>\n</math> and angles <math alttext=\"upper C\"><mi>C</mi>\n</math> and <math alttext=\"upper F\"><mi>F</mi>\n</math> are right angles. In similar triangles, the tangents of corresponding angles are equal. Therefore, if <math alttext=\"tangent upper A equals StartFraction 21 Over 20 EndFraction\"><mi>tan</mi><mi>A</mi><mo>=</mo><mfrac><mn>21</mn><mn>20</mn></mfrac></math>, then&nbsp;<math alttext=\"tangent upper D equals StartFraction 21 Over 20 EndFraction\"><mi>tan</mi><mi>D</mi><mo>=</mo><mfrac><mn>21</mn><mn>20</mn></mfrac></math>. In a right triangle, the tangent of an acute angle is the ratio of the length of the leg opposite the angle to the length of the leg adjacent to the angle. Therefore, in triangle <math alttext=\"upper D upper E upper F\"><mrow>\n\t<mi>D</mi>\n\t<mi>E</mi>\n\t<mi>F</mi>\n</mrow>\n</math>, if <math alttext=\"tangent upper D equals StartFraction 21 Over 20 EndFraction\"><mi>tan</mi><mi>D</mi><mo>=</mo><mfrac><mn>21</mn><mn>20</mn></mfrac></math>, the ratio of the length of <math alttext=\"line segment upper E upper F\"><mover><mrow><mi>E</mi><mi>F</mi></mrow><mo>¯</mo></mover></math> to the length of&nbsp;<math alttext=\"line segment upper D upper F\"><mover><mrow><mi>D</mi><mi>F</mi></mrow><mo>¯</mo></mover></math> is <math alttext=\"StartFraction 21 Over 20 EndFraction\"><mfrac>\n\t<mn>21</mn>\n\t<mn>20</mn>\n</mfrac>\n</math>. If the lengths of <math alttext=\"line segment upper E upper F\"><mover><mrow><mi>E</mi><mi>F</mi></mrow><mo>¯</mo></mover></math> and&nbsp;<math alttext=\"line segment upper D upper F\"><mover><mrow><mi>D</mi><mi>F</mi></mrow><mo>¯</mo></mover></math> are <math alttext=\"21\"><mn>21</mn>\n</math> and <math alttext=\"20\"><mn>20</mn>\n</math>, respectively, then the ratio of the length of&nbsp;<math alttext=\"line segment upper E upper F\"><mover><mrow><mi>E</mi><mi>F</mi></mrow><mo>¯</mo></mover></math> to the length of&nbsp;<math alttext=\"line segment upper D upper F\"><mover><mrow><mi>D</mi><mi>F</mi></mrow><mo>¯</mo></mover></math> is <math alttext=\"StartFraction 21 Over 20 EndFraction\"><mfrac>\n\t<mn>21</mn>\n\t<mn>20</mn>\n</mfrac>\n</math>. In a right triangle, the sine of an acute angle is the ratio of the length of the leg opposite the angle to the length of the hypotenuse. Therefore, the value of <math alttext=\"sine upper D\"><mi>sin</mi><mi>D</mi></math> is the ratio of the length of&nbsp;<math alttext=\"line segment upper E upper F\"><mover><mrow><mi>E</mi><mi>F</mi></mrow><mo>¯</mo></mover></math> to the length of&nbsp;<math alttext=\"line segment upper D upper E\"><mover><mrow><mi>D</mi><mi>E</mi></mrow><mo>¯</mo></mover></math>. The length of&nbsp;<math alttext=\"line segment upper D upper E\"><mover><mrow><mi>D</mi><mi>E</mi></mrow><mo>¯</mo></mover></math> can be calculated using the Pythagorean theorem, which states that if the lengths of the legs of a right triangle are <math alttext=\"a\"><mi>a</mi>\n</math> and <math alttext=\"b\"><mi>b</mi>\n</math> and the length of the hypotenuse is <math alttext=\"c\"><mi>c</mi>\n</math>, then <math alttext=\"a squared plus b squared equals c squared\"><mrow>\n\t<mrow>\n\t\t<msup>\n\t\t\t<mi>a</mi>\n\t\t\t<mn>2</mn>\n\t\t</msup>\n\t\t<mo>+</mo>\n\t\t<msup>\n\t\t\t<mi>b</mi>\n\t\t\t<mn>2</mn>\n\t\t</msup>\n\t</mrow>\n\t<mo>=</mo>\n\t<msup>\n\t\t<mi>c</mi>\n\t\t<mn>2</mn>\n\t</msup>\n</mrow>\n</math>. Therefore, if the lengths of <math alttext=\"line segment upper E upper F\"><mover><mrow><mi>E</mi><mi>F</mi></mrow><mo>¯</mo></mover></math> and&nbsp;<math alttext=\"line segment upper D upper F\"><mover><mrow><mi>D</mi><mi>F</mi></mrow><mo>¯</mo></mover></math> are <math alttext=\"21\"><mn>21</mn>\n</math> and <math alttext=\"20\"><mn>20</mn>\n</math>, respectively, then <math alttext=\"left parenthesis 21 right parenthesis squared plus left parenthesis 20 right parenthesis squared equals left parenthesis upper D upper E right parenthesis squared\"><msup><mfenced><mn>21</mn></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mn>20</mn></mfenced><mn>2</mn></msup><mo>=</mo><msup><mfenced><mrow><mi>D</mi><mi>E</mi></mrow></mfenced><mn>2</mn></msup></math>, or&nbsp;<math alttext=\"841 equals left parenthesis upper D upper E right parenthesis squared\"><mn>841</mn><mo>=</mo><msup><mfenced><mrow><mi>D</mi><mi>E</mi></mrow></mfenced><mn>2</mn></msup></math>. Taking the positive square root of both sides of this equation yields&nbsp;<math alttext=\"29 equals upper D upper E\"><mn>29</mn><mo>=</mo><mi>D</mi><mi>E</mi></math>. Therefore, if the lengths of&nbsp;<math alttext=\"line segment upper E upper F\"><mover><mrow><mi>E</mi><mi>F</mi></mrow><mo>¯</mo></mover></math> and&nbsp;<math alttext=\"line segment upper D upper F\"><mover><mrow><mi>D</mi><mi>F</mi></mrow><mo>¯</mo></mover></math> are <math alttext=\"21\"><mn>21</mn>\n</math> and <math alttext=\"20\"><mn>20</mn>\n</math>, respectively, then the length of <math alttext=\"line segment upper D upper E\"><mover><mrow><mi>D</mi><mi>E</mi></mrow><mo>¯</mo></mover></math> is <math alttext=\"29\"><mn>29</mn>\n</math> and the ratio of the length of <math alttext=\"line segment upper E upper F\"><mover><mrow><mi>E</mi><mi>F</mi></mrow><mo>¯</mo></mover></math> to the length of <math alttext=\"line segment upper D upper E\"><mover><mrow><mi>D</mi><mi>E</mi></mrow><mo>¯</mo></mover></math> is&nbsp;<math alttext=\"StartFraction 21 Over 29 EndFraction\"><mfrac>\n\t<mn>21</mn>\n\t<mn>29</mn>\n</mfrac>\n</math>. Thus, if <math alttext=\"tangent upper A equals StartFraction 21 Over 20 EndFraction\"><mi>tan</mi><mi>A</mi><mo>=</mo><mfrac><mn>21</mn><mn>20</mn></mfrac></math>, the value of&nbsp;<math alttext=\"sine upper D\"><mi>sin</mi><mi>D</mi></math> is <math alttext=\"StartFraction 21 Over 29 EndFraction\"><mfrac>\n\t<mn>21</mn>\n\t<mn>29</mn>\n</mfrac>\n</math>. Note that 21/29, .7241, and 0.724 are examples of ways to enter a correct answer.</p>","correct_answer":[".7241","21/29"],"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":[],"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}}