{"id":"cb_question_bank:dcb161a3-c40c-4077-bb00-4eb0e55d5e76","source_id":"cb_question_bank","source_item_id":"dcb161a3-c40c-4077-bb00-4eb0e55d5e76","external_id":"13cc44bf-f2ac-4f02-ac05-440e4b894f96","ibn":null,"question_id":"8027db3f","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":"In triangle upper J upper K upper L , cosine left parenthesis upper K right parenthesis equals StartFraction 24 Over 51 EndFraction and angle upper J is a right angle. What is the value of cosine left parenthesis upper L right parenthesis ?","stimulus":"","stimulus_html":null,"stem_html":"<p>In triangle <math alttext=\"upper J upper K upper L\"><mrow>\n\t<mi>J</mi>\n\t<mi>K</mi>\n\t<mi>L</mi>\n</mrow>\n</math>, <math alttext=\"cosine left parenthesis upper K right parenthesis equals StartFraction 24 Over 51 EndFraction\"><mrow>\n\t<mi>cos</mi>\n\t<mfenced>\n\t\t<mi>K</mi>\n\t</mfenced>\n</mrow>\n<mo>=</mo><mfrac><mrow><mn>24</mn></mrow><mrow><mn>51</mn></mrow></mfrac></math> and angle <math alttext=\"upper J\"><mi>J</mi>\n</math> is a right angle. What is the value of <math alttext=\"cosine left parenthesis upper L right parenthesis\"><mrow>\n\t<mi>cos</mi>\n\t<mfenced>\n\t\t<mi>L</mi>\n\t</mfenced>\n</mrow>\n</math>?</p>","rationale_html":"<p style=\"text-align: left;\">The correct answer is <math alttext=\"StartFraction 15 Over 17 EndFraction\"><mfrac><mn>15</mn><mn>17</mn></mfrac></math>. It's given that angle <math alttext=\"upper J\"><mi>J</mi>\n</math> is the right angle in triangle <math alttext=\"upper J upper K upper L\"><mi>J</mi><mi>K</mi><mi>L</mi></math>. Therefore, the acute angles of triangle&nbsp;<math alttext=\"upper J upper K upper L\"><mi>J</mi><mi>K</mi><mi>L</mi></math> are angle <math alttext=\"upper K\"><mi>K</mi>\n</math> and angle <math alttext=\"upper L\"><mi>L</mi>\n</math>. The hypotenuse of a right triangle is the side opposite its right angle. Therefore, the hypotenuse of triangle <math alttext=\"upper J upper K upper L\"><mi>J</mi><mi>K</mi><mi>L</mi></math> is side <math alttext=\"upper K upper L\"><mi>K</mi><mi>L</mi></math>. The cosine of an acute angle in a right triangle is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse. It's given that <math alttext=\"cosine left parenthesis upper K right parenthesis equals StartFraction 24 Over 51 EndFraction\"><mi>cos</mi><mfenced><mi>K</mi></mfenced><mo>=</mo><mfrac><mn>24</mn><mn>51</mn></mfrac></math>. This can be written as&nbsp;<math alttext=\"cosine left parenthesis upper K right parenthesis equals eight seventeenths\"><mi>cos</mi><mfenced><mi>K</mi></mfenced><mo>=</mo><mfrac><mn>8</mn><mn>17</mn></mfrac></math>. Since the cosine of angle <math alttext=\"upper K\"><mi>K</mi>\n</math> is a ratio, it follows that the length of the side adjacent to angle <math alttext=\"upper K\"><mi>K</mi>\n</math> is <math alttext=\"8 n\"><mrow>\n\t<mn>8</mn>\n\t<mi>n</mi>\n</mrow>\n</math> and the length of the hypotenuse is <math alttext=\"17 n\"><mrow>\n\t<mn>17</mn>\n\t<mi>n</mi>\n</mrow>\n</math>, where <math alttext=\"n\"><mi>n</mi>\n</math> is a constant. Therefore, <math alttext=\"upper J upper K equals 8 n\"><mi>J</mi><mi>K</mi><mo>=</mo><mn>8</mn><mi>n</mi></math> and&nbsp;<math alttext=\"upper K upper L equals 17 n\"><mi>K</mi><mi>L</mi><mo>=</mo><mn>17</mn><mi>n</mi></math>. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. For triangle&nbsp;<math alttext=\"upper J upper K upper L\"><mi>J</mi><mi>K</mi><mi>L</mi></math>, it follows that&nbsp;<math alttext=\"left parenthesis upper J upper K right parenthesis squared plus left parenthesis upper J upper L right parenthesis squared equals left parenthesis upper K upper L right parenthesis squared\"><msup><mfenced><mrow><mi>J</mi><mi>K</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>J</mi><mi>L</mi></mrow></mfenced><mn>2</mn></msup><mo>=</mo><msup><mfenced><mrow><mi>K</mi><mi>L</mi></mrow></mfenced><mn>2</mn></msup></math>. Substituting <math alttext=\"8 n\"><mrow>\n\t<mn>8</mn>\n\t<mi>n</mi>\n</mrow>\n</math> for <math alttext=\"upper J upper K\"><mrow>\n\t<mi>J</mi>\n\t<mi>K</mi>\n</mrow>\n</math> and <math alttext=\"17 n\"><mrow>\n\t<mn>17</mn>\n\t<mi>n</mi>\n</mrow>\n</math> for <math alttext=\"upper K upper L\"><mrow>\n\t<mi>K</mi>\n\t<mi>L</mi>\n</mrow>\n</math> yields <math alttext=\"left parenthesis 8 n right parenthesis squared plus left parenthesis upper J upper L right parenthesis squared equals left parenthesis 17 n right parenthesis squared\"><msup><mfenced><mrow><mn>8</mn><mi>n</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>J</mi><mi>L</mi></mrow></mfenced><mn>2</mn></msup><mo>=</mo><msup><mfenced><mrow><mn>17</mn><mi>n</mi></mrow></mfenced><mn>2</mn></msup></math>. This is equivalent to&nbsp;<math alttext=\"64 n squared plus left parenthesis upper J upper L right parenthesis squared equals 289 n squared\"><mn>64</mn><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>J</mi><mi>L</mi></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>289</mn><msup><mi>n</mi><mn>2</mn></msup></math>. Subtracting&nbsp;<math alttext=\"64 n squared\"><mn>64</mn><msup><mi>n</mi><mn>2</mn></msup></math> from each side of this equation yields <math alttext=\"left parenthesis upper J upper L right parenthesis squared equals 225 n squared\"><msup><mfenced><mrow><mi>J</mi><mi>L</mi></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>225</mn><msup><mi>n</mi><mn>2</mn></msup></math>. Taking the square root of each side of this equation yields&nbsp;<math alttext=\"upper J upper L equals 15 n\"><mi>J</mi><mi>L</mi><mo>=</mo><mn>15</mn><mi>n</mi></math>. Since&nbsp;<math alttext=\"cosine left parenthesis upper L right parenthesis equals StartFraction upper J upper L Over upper K upper L EndFraction\"><mi>cos</mi><mfenced><mi>L</mi></mfenced><mo>=</mo><mfrac><mrow><mi>J</mi><mi>L</mi></mrow><mrow><mi>K</mi><mi>L</mi></mrow></mfrac></math>, it follows that <math alttext=\"cosine left parenthesis upper L right parenthesis equals StartFraction 15 n Over 17 n EndFraction\"><mi>cos</mi><mfenced><mi>L</mi></mfenced><mo>=</mo><mfrac><mrow><mn>15</mn><mi>n</mi></mrow><mrow><mn>17</mn><mi>n</mi></mrow></mfrac></math>, which can be rewritten as <math alttext=\"cosine left parenthesis upper L right parenthesis equals StartFraction 15 Over 17 EndFraction\"><mi>cos</mi><mfenced><mi>L</mi></mfenced><mo>=</mo><mfrac><mn>15</mn><mn>17</mn></mfrac></math>. Note that 15/17, .8824, .8823, and 0.882 are examples of ways to enter a correct answer.</p>","correct_answer":[".8823",".8824","15/17"],"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}}