{"id":"cb_question_bank:d5f36b76-e82c-4053-94a5-25c3b947e15d","source_id":"cb_question_bank","source_item_id":"d5f36b76-e82c-4053-94a5-25c3b947e15d","external_id":"ddfe8011-1fec-4e7d-81bb-e842017e4525","ibn":null,"question_id":"2be01bd9","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":"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 angle upper C corresponds to angle upper F . Angles upper C and upper F are right angles. If tangent left parenthesis upper A right parenthesis equals StartFraction 50 Over 7 EndFraction , what is the value of tangent left parenthesis upper E right parenthesis ?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: left;\">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 angle <math alttext=\"upper C\"><mi>C</mi>\n</math> corresponds to angle <math alttext=\"upper F\"><mi>F</mi>\n</math>. &nbsp;Angles <math alttext=\"upper C\"><mi>C</mi>\n</math> and <math alttext=\"upper F\"><mi>F</mi>\n</math> are right angles. If <math alttext=\"tangent left parenthesis upper A right parenthesis equals StartFraction 50 Over 7 EndFraction\"><mrow>\n\t<mrow>\n\t\t<mi>tan</mi>\n\t\t<mfenced>\n\t\t\t<mi>A</mi>\n\t\t</mfenced>\n\t</mrow>\n\t<mo>=</mo>\n\t<mfrac>\n\t\t<mn>50</mn>\n\t\t<mn>7</mn>\n\t</mfrac>\n</mrow>\n</math>, what is the value of&nbsp; <math alttext=\"tangent left parenthesis upper E right parenthesis\"><mrow>\n\t<mi>tan</mi>\n\t<mfenced>\n\t\t<mi>E</mi>\n\t</mfenced>\n</mrow>\n</math>?</p>","rationale_html":"<p style=\"text-align: left;\">The correct answer is <math alttext=\"seven fiftieths\"><mfrac>\n\t<mn>7</mn>\n\t<mn>50</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 angle <math alttext=\"upper C\"><mi>C</mi>\n</math> corresponds to angle <math alttext=\"upper F\"><mi>F</mi>\n</math>. In similar triangles, the tangents of corresponding angles are equal. Since angle <math alttext=\"upper A\"><mi>A</mi>\n</math> and angle <math alttext=\"upper D\"><mi>D</mi>\n</math> are corresponding angles, if <math alttext=\"tangent left parenthesis upper A right parenthesis equals StartFraction 50 Over 7 EndFraction\"><mi>tan</mi><mfenced><mi>A</mi></mfenced><mo>=</mo><mfrac><mn>50</mn><mn>7</mn></mfrac></math>, then&nbsp;<math alttext=\"tangent left parenthesis upper D right parenthesis equals StartFraction 50 Over 7 EndFraction\"><mi>tan</mi><mfenced><mi>D</mi></mfenced><mo>=</mo><mfrac><mn>50</mn><mn>7</mn></mfrac></math>. It's also given that angles <math alttext=\"upper C\"><mi>C</mi>\n</math> and <math alttext=\"upper F\"><mi>F</mi>\n</math> are right angles. It follows that 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> is a right triangle with acute angles <math alttext=\"upper D\"><mi>D</mi>\n</math> and <math alttext=\"upper E\"><mi>E</mi>\n</math>. The tangent of one acute angle in a right triangle is the inverse of the tangent of the other acute angle in the triangle. Therefore, <math alttext=\"tangent left parenthesis upper E right parenthesis equals StartFraction 1 Over tangent left parenthesis upper D right parenthesis EndFraction\"><mi>tan</mi><mfenced><mi>E</mi></mfenced><mo>=</mo><mfrac><mn>1</mn><mrow><mi>tan</mi><mfenced><mi>D</mi></mfenced></mrow></mfrac></math>. Substituting <math alttext=\"StartFraction 50 Over 7 EndFraction\"><mfrac>\n\t<mn>50</mn>\n\t<mn>7</mn>\n</mfrac>\n</math> for&nbsp;<math alttext=\"tangent left parenthesis upper D right parenthesis\"><mi>tan</mi><mfenced><mi>D</mi></mfenced></math> in this equation yields&nbsp;<math alttext=\"tangent left parenthesis upper E right parenthesis equals StartStartFraction 1 OverOver StartFraction 50 Over 7 EndFraction EndEndFraction\"><mi>tan</mi><mfenced><mi>E</mi></mfenced><mo>=</mo><mfrac><mn>1</mn><mstyle displaystyle=\"true\"><mfrac><mn>50</mn><mn>7</mn></mfrac></mstyle></mfrac></math>, or&nbsp;<math alttext=\"tangent left parenthesis upper E right parenthesis equals seven fiftieths\"><mi>tan</mi><mfenced><mi>E</mi></mfenced><mo>=</mo><mfrac><mn>7</mn><mn>50</mn></mfrac></math>. Thus, if&nbsp;<math alttext=\"tangent left parenthesis upper A right parenthesis equals StartFraction 50 Over 7 EndFraction\"><mi>tan</mi><mfenced><mi>A</mi></mfenced><mo>=</mo><mfrac><mn>50</mn><mn>7</mn></mfrac></math>, the value of&nbsp;<math alttext=\"tangent left parenthesis upper E right parenthesis\"><mi>tan</mi><mfenced><mi>E</mi></mfenced></math> is <math alttext=\"seven fiftieths\"><mfrac>\n\t<mn>7</mn>\n\t<mn>50</mn>\n</mfrac>\n</math>. Note that 7/50 and .14 are examples of ways to enter a correct answer.</p>","correct_answer":[".14","7/50"],"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}}