{"id":"cb_question_bank:bf37bde6-6a48-49e9-ab8c-98f6afe5f1d6","source_id":"cb_question_bank","source_item_id":"bf37bde6-6a48-49e9-ab8c-98f6afe5f1d6","external_id":"ef234bbf-719d-4bc2-a841-937a532d135b","ibn":null,"question_id":"a5aee181","program":"SAT","module":"math","domain_code":"S","domain":"Geometry and Trigonometry","skill_code":"S.C.","skill":"Right triangles and trigonometry","difficulty":"M","score_band":5,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"The length of a rectangle’s diagonal is 5 17 , and the length of the rectangle’s shorter side is 5 . What is the length of the rectangle’s longer side?","stimulus":"","stimulus_html":null,"stem_html":"<p>The length of a rectangle’s diagonal is <math><mrow>\n\t<mn>5</mn>\n\t<msqrt>\n\t\t<mn>17</mn>\n\t</msqrt>\n</mrow>\n</math>, and the length of the rectangle’s shorter side is <math><mn>5</mn>\n</math>. What is the length of the rectangle’s longer side?</p>","rationale_html":"<p>Choice B is correct. A rectangle’s diagonal divides a rectangle into two congruent right triangles, where the diagonal is the hypotenuse of both triangles. It’s given that the length of the diagonal is&nbsp;<math alttext=\"5 StartRoot 17 EndRoot\"><mn>5</mn><msqrt><mn>17</mn></msqrt></math> and the length of the rectangle’s shorter side is <math alttext=\"5\"><mn>5</mn>\n</math>. Therefore, each of the two right triangles formed by the rectangle’s diagonal has a hypotenuse with length <math alttext=\"5 StartRoot 17 EndRoot\"><mn>5</mn><msqrt><mn>17</mn></msqrt></math>, and a shorter leg with length <math alttext=\"5\"><mn>5</mn>\n</math>. To calculate the length of the longer leg of each right triangle, the Pythagorean theorem, <math alttext=\"a squared plus b squared equals c squared\"><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo>=</mo><msup><mi>c</mi><mn>2</mn></msup></math>, can be used, where&nbsp;<math alttext=\"a\"><mi>a</mi></math> and&nbsp;<math alttext=\"b\"><mi>b</mi></math> are the lengths of the legs and&nbsp;<math alttext=\"c\"><mi>c</mi></math> is the length of the hypotenuse of the triangle. Substituting <math alttext=\"5\"><mn>5</mn>\n</math> for&nbsp;<math alttext=\"a\"><mi>a</mi></math> and <math alttext=\"5 StartRoot 17 EndRoot\"><mn>5</mn><msqrt><mn>17</mn></msqrt></math> for&nbsp;<math alttext=\"c\"><mi>c</mi></math> in the equation <math alttext=\"a squared plus b squared equals c squared\"><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo>=</mo><msup><mi>c</mi><mn>2</mn></msup></math> &nbsp;yields <math alttext=\"5 squared plus b squared equals left parenthesis 5 StartRoot 17 EndRoot right parenthesis squared\"><msup><mn>5</mn><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo>=</mo><msup><mfenced><mrow><mn>5</mn><msqrt><mn>17</mn></msqrt></mrow></mfenced><mn>2</mn></msup></math>, which is equivalent to <math alttext=\"25 plus b squared equals 25 left parenthesis 17 right parenthesis\"><mn>25</mn><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo>=</mo><mn>25</mn><mfenced><mn>17</mn></mfenced></math>, or <math alttext=\"25 plus b squared equals 425\"><mn>25</mn><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo>=</mo><mn>425</mn></math>. Subtracting <math alttext=\"25\"><mn>25</mn>\n</math> from each side of this equation yields <math alttext=\"b squared equals 400\"><msup><mi>b</mi><mn>2</mn></msup><mo>=</mo><mn>400</mn></math>. Taking the positive square root of each side of this equation yields <math alttext=\"b equals 20\"><mi>b</mi><mo>=</mo><mn>20</mn></math>. Therefore, the length of the longer leg of each right triangle formed by the diagonal of the rectangle is <math alttext=\"20\"><mn>20</mn>\n</math>. It follows that the length of the rectangle’s longer side is <math alttext=\"20\"><mn>20</mn>\n</math>.</p>\n<p>Choice A is incorrect and may result from dividing the length of the rectangle’s diagonal by the length of the rectangle’s shorter side, rather than substituting these values into the Pythagorean theorem.</p>\n<p>Choice C is incorrect and may result from using the length of the rectangle’s diagonal as the length of a leg of the right triangle, rather than the length of the hypotenuse.</p>\n<p>Choice D is incorrect. This is the square of the length of the rectangle’s longer side.</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><msqrt>\n\t<mn>17</mn>\n</msqrt>\n</math></p>","ord":0},{"label":"B","content_html":"<p><math><mn>20</mn>\n</math></p>","ord":1},{"label":"C","content_html":"<p><math><mrow>\n\t<mn>15</mn>\n\t<msqrt>\n\t\t<mn>2</mn>\n\t</msqrt>\n</mrow>\n</math></p>","ord":2},{"label":"D","content_html":"<p><math><mn>400</mn>\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}}