{"id":"cb_question_bank:a4f4a4da-c7a5-418d-8aab-3fc96ce65eb9","source_id":"cb_question_bank","source_item_id":"a4f4a4da-c7a5-418d-8aab-3fc96ce65eb9","external_id":"21c2c201-137b-4fd3-9fa4-520c318b5319","ibn":null,"question_id":"5b4757df","program":"SAT","module":"math","domain_code":"S","domain":"Geometry and Trigonometry","skill_code":"S.B.","skill":"Lines, angles, and triangles","difficulty":"H","score_band":7,"answer_type":"spr","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"In triangle upper R upper S upper T , angle upper T is a right angle, point upper L lies on line segment upper R upper S , point upper K lies on line segment upper S upper T , and line segment upper L upper K is parallel to line segment upper R upper T . If the length of line segment upper R upper T is 72 units, the length of line segment upper L upper K is 24 units, and the area of triangle upper R upper S upper T is 792 square units, what is the length of line segment upper K upper T , in units?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: left;\">In triangle&nbsp;<math alttext=\"upper R upper S upper T\"><mi>R</mi><mi>S</mi><mi>T</mi></math>, angle <math alttext=\"upper T\"><mi>T</mi>\n</math> is a right angle, point <math alttext=\"upper L\"><mi>L</mi>\n</math> lies on <math alttext=\"line segment upper R upper S\"><mover><mrow><mi>R</mi><mi>S</mi></mrow><mo>¯</mo></mover></math>, point <math alttext=\"upper K\"><mi>K</mi>\n</math> lies on <math alttext=\"line segment upper S upper T\"><mover><mrow><mi>S</mi><mi>T</mi></mrow><mo>¯</mo></mover></math>, and <math alttext=\"line segment upper L upper K\"><mover><mrow><mi>L</mi><mi>K</mi></mrow><mo>¯</mo></mover></math> is parallel to <math alttext=\"line segment upper R upper T\"><mover><mrow><mi>R</mi><mi>T</mi></mrow><mo>¯</mo></mover></math>. If the length of&nbsp;<math alttext=\"line segment upper R upper T\"><mover><mrow><mi>R</mi><mi>T</mi></mrow><mo>¯</mo></mover></math> is <math alttext=\"72\"><mn>72</mn>\n</math> units, the length of <math alttext=\"line segment upper L upper K\"><mover><mrow><mi>L</mi><mi>K</mi></mrow><mo>¯</mo></mover></math> is <math alttext=\"24\"><mn>24</mn>\n</math> units, and the area of triangle <math alttext=\"upper R upper S upper T\"><mi>R</mi><mi>S</mi><mi>T</mi></math> is <math alttext=\"792\"><mn>792</mn>\n</math> square units, what is the length of <math alttext=\"line segment upper K upper T\"><mover><mrow><mi>K</mi><mi>T</mi></mrow><mo>¯</mo></mover></math>, in units?</p>","rationale_html":"<p style=\"text-align: left;\">The correct answer is <math alttext=\"StartFraction 44 Over 3 EndFraction\"><mfrac>\n\t<mn>44</mn>\n\t<mn>3</mn>\n</mfrac>\n</math>. It's given that in 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>, angle <math alttext=\"upper T\"><mi>T</mi>\n</math> is a right angle. The area of a right triangle can be found using the formula <math alttext=\"upper A equals one half script l 1 script l 2\"><mi>A</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>l</mi><mn>1</mn></msub><msub><mi>l</mi><mn>2</mn></msub></math>, where <math alttext=\"upper A\"><mi>A</mi>\n</math> represents the area of the right triangle, <math alttext=\"script l 1\"><msub><mi>l</mi><mn>1</mn></msub></math> represents the length of one leg of the triangle, and <math alttext=\"script l 2\"><msub><mi>l</mi><mn>2</mn></msub></math> represents the length of the other leg of the triangle. In 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>, the two legs are <math alttext=\"line segment upper R upper T\"><mover><mrow><mi>R</mi><mi>T</mi></mrow><mo>¯</mo></mover></math> and <math alttext=\"line segment upper S upper T\"><mover><mrow><mi>S</mi><mi>T</mi></mrow><mo>¯</mo></mover></math>. Therefore, if the length of <math alttext=\"line segment upper R upper T\"><mover><mrow><mi>R</mi><mi>T</mi></mrow><mo>¯</mo></mover></math> is <math alttext=\"72\"><mn>72</mn>\n</math> and the area of triangle <math alttext=\"upper R upper S upper T\"><mi>R</mi><mi>S</mi><mi>T</mi></math> is <math alttext=\"792\"><mn>792</mn>\n</math>, then <math alttext=\"792 equals one half left parenthesis 72 right parenthesis left parenthesis upper S upper T right parenthesis\"><mn>792</mn><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced><mn>72</mn></mfenced><mfenced><mrow><mi>S</mi><mi>T</mi></mrow></mfenced></math>, or&nbsp;<math alttext=\"792 equals left parenthesis 36 right parenthesis left parenthesis upper S upper T right parenthesis\"><mn>792</mn><mo>=</mo><mfenced><mn>36</mn></mfenced><mfenced><mrow><mi>S</mi><mi>T</mi></mrow></mfenced></math>. Dividing both sides of this equation by <math alttext=\"36\"><mn>36</mn>\n</math> yields&nbsp;<math alttext=\"22 equals upper S upper T\"><mn>22</mn><mo>=</mo><mi>S</mi><mi>T</mi></math>. Therefore, the length of <math alttext=\"line segment upper S upper T\"><mover><mrow><mi>S</mi><mi>T</mi></mrow><mo>¯</mo></mover></math> is <math alttext=\"22\"><mn>22</mn>\n</math>. It's also given that point <math alttext=\"upper L\"><mi>L</mi>\n</math> lies on <math alttext=\"line segment upper R upper S\"><mover><mrow><mi>R</mi><mi>S</mi></mrow><mo>¯</mo></mover></math>, point <math alttext=\"upper K\"><mi>K</mi>\n</math> lies on&nbsp;<math alttext=\"line segment upper S upper T\"><mover><mrow><mi>S</mi><mi>T</mi></mrow><mo>¯</mo></mover></math>, and&nbsp;<math alttext=\"line segment upper L upper K\"><mover><mrow><mi>L</mi><mi>K</mi></mrow><mo>¯</mo></mover></math> is parallel to&nbsp;<math alttext=\"line segment upper R upper T\"><mover><mrow><mi>R</mi><mi>T</mi></mrow><mo>¯</mo></mover></math>. It follows that angle <math alttext=\"upper L upper K upper S\"><mi>L</mi><mi>K</mi><mi>S</mi></math> is a right angle. Since triangles <math alttext=\"upper R upper S upper T\"><mi>R</mi><mi>S</mi><mi>T</mi></math> and <math alttext=\"upper L upper S upper K\"><mi>L</mi><mi>S</mi><mi>K</mi></math> share angle <math alttext=\"upper S\"><mi>S</mi>\n</math> and have right angles <math alttext=\"upper T\"><mi>T</mi>\n</math> and <math alttext=\"upper K\"><mi>K</mi>\n</math>, respectively, triangles <math alttext=\"upper R upper S upper T\"><mi>R</mi><mi>S</mi><mi>T</mi></math> and&nbsp;<math alttext=\"upper L upper S upper K\"><mi>L</mi><mi>S</mi><mi>K</mi></math> are similar triangles. Therefore, the ratio of the length of&nbsp;<math alttext=\"line segment upper R upper T\"><mover><mrow><mi>R</mi><mi>T</mi></mrow><mo>¯</mo></mover></math> to the length of&nbsp;<math alttext=\"line segment upper L upper K\"><mover><mrow><mi>L</mi><mi>K</mi></mrow><mo>¯</mo></mover></math> is equal to the ratio of the length of&nbsp;<math alttext=\"line segment upper S upper T\"><mover><mrow><mi>S</mi><mi>T</mi></mrow><mo>¯</mo></mover></math> to the length of&nbsp;<math alttext=\"line segment upper S upper K\"><mover><mrow><mi>S</mi><mi>K</mi></mrow><mo>¯</mo></mover></math>. If the length of&nbsp;<math alttext=\"line segment upper R upper T\"><mover><mrow><mi>R</mi><mi>T</mi></mrow><mo>¯</mo></mover></math> is <math alttext=\"72\"><mn>72</mn>\n</math> and the length of&nbsp;<math alttext=\"line segment upper L upper K\"><mover><mrow><mi>L</mi><mi>K</mi></mrow><mo>¯</mo></mover></math> is <math alttext=\"24\"><mn>24</mn>\n</math>, it follows that the ratio of the length of&nbsp;<math alttext=\"line segment upper R upper T\"><mover><mrow><mi>R</mi><mi>T</mi></mrow><mo>¯</mo></mover></math> to the length of&nbsp;<math alttext=\"line segment upper L upper K\"><mover><mrow><mi>L</mi><mi>K</mi></mrow><mo>¯</mo></mover></math> is&nbsp;<math alttext=\"StartFraction 72 Over 24 EndFraction\"><mfrac><mn>72</mn><mn>24</mn></mfrac></math>, or <math alttext=\"3\"><mn>3</mn>\n</math>, so the ratio of the length of&nbsp;<math alttext=\"line segment upper S upper T\"><mover><mrow><mi>S</mi><mi>T</mi></mrow><mo>¯</mo></mover></math> to the length of&nbsp;<math alttext=\"line segment upper S upper K\"><mover><mrow><mi>S</mi><mi>K</mi></mrow><mo>¯</mo></mover></math> is <math alttext=\"3\"><mn>3</mn>\n</math>. Therefore, <math alttext=\"StartFraction 22 Over upper S upper K EndFraction equals 3\"><mfrac><mn>22</mn><mrow><mi>S</mi><mi>K</mi></mrow></mfrac><mo>=</mo><mn>3</mn></math>. Multiplying both sides of this equation by&nbsp;<math alttext=\"upper S upper K\"><mi>S</mi><mi>K</mi></math> yields&nbsp;<math alttext=\"22 equals left parenthesis 3 right parenthesis left parenthesis upper S upper K right parenthesis\"><mn>22</mn><mo>=</mo><mfenced><mn>3</mn></mfenced><mfenced><mrow><mi>S</mi><mi>K</mi></mrow></mfenced></math>. Dividing both sides of this equation by <math alttext=\"3\"><mn>3</mn>\n</math> yields&nbsp;<math alttext=\"StartFraction 22 Over 3 EndFraction equals upper S upper K\"><mfrac><mn>22</mn><mn>3</mn></mfrac><mo>=</mo><mi>S</mi><mi>K</mi></math>. Since the length of <math alttext=\"line segment upper S upper T\"><mover><mrow><mi>S</mi><mi>T</mi></mrow><mo>¯</mo></mover></math>, <math alttext=\"22\"><mn>22</mn>\n</math>, is the sum of the length of <math alttext=\"line segment upper S upper K\"><mover><mrow><mi>S</mi><mi>K</mi></mrow><mo>¯</mo></mover></math>, <math alttext=\"StartFraction 22 Over 3 EndFraction\"><mfrac>\n\t<mn>22</mn>\n\t<mn>3</mn>\n</mfrac>\n</math>, and the length of <math alttext=\"line segment upper K upper T\"><mover><mrow><mi>K</mi><mi>T</mi></mrow><mo>¯</mo></mover></math>, it follows that the length of&nbsp;<math alttext=\"line segment upper K upper T\"><mover><mrow><mi>K</mi><mi>T</mi></mrow><mo>¯</mo></mover></math> is&nbsp;<math alttext=\"22 minus StartFraction 22 Over 3 EndFraction\"><mn>22</mn><mo>-</mo><mfrac><mn>22</mn><mn>3</mn></mfrac></math>, or <math alttext=\"StartFraction 44 Over 3 EndFraction\"><mfrac>\n\t<mn>44</mn>\n\t<mn>3</mn>\n</mfrac>\n</math>. Note that 44/3, 14.66, and 14.67 are examples of ways to enter a correct answer.</p>","correct_answer":["14.66","14.67","44/3"],"has_media":false,"has_mathml":true,"has_table":false,"source_created_at":"2023-08-02T20:25:59.838000+00:00","source_updated_at":"2023-08-02T20:25:59.838000+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}}