{"id":"cb_question_bank:b5bcc30b-9e5e-4dba-b348-a589fe3c479c","source_id":"cb_question_bank","source_item_id":"b5bcc30b-9e5e-4dba-b348-a589fe3c479c","external_id":"575135d8-2a82-4346-993f-f5d9b5921696","ibn":null,"question_id":"7c25b0dc","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":"The length of a rectangle’s diagonal is 3 StartRoot 17 EndRoot , and the length of the rectangle’s shorter side is 3 . What is the length of the rectangle’s longer side?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: left;\">The length of a rectangle’s diagonal is <math alttext=\"3 StartRoot 17 EndRoot\"><mrow>\n\t<mn>3</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 alttext=\"3\"><mn>3</mn>\n</math>. What is the length of the rectangle’s longer side?</p>","rationale_html":"<p style=\"text-align: left;\">The correct answer is <math alttext=\"12\"><mn>12</mn>\n</math>. The diagonal of a rectangle forms a right triangle, where the shorter side and the longer side of the rectangle are the legs of the triangle and the diagonal of the rectangle is the hypotenuse of the triangle. It's given that the length of the rectangle's diagonal is <math alttext=\"3 StartRoot 17 EndRoot\"><mn>3</mn><msqrt><mn>17</mn></msqrt></math> and the length of the rectangle's shorter side is <math alttext=\"3\"><mn>3</mn>\n</math>. Thus, the length of the hypotenuse of the right triangle formed by the diagonal is <math alttext=\"3 StartRoot 17 EndRoot\"><mn>3</mn><msqrt><mn>17</mn></msqrt></math> and the length of one of the legs is <math alttext=\"3\"><mn>3</mn>\n</math>. By the Pythagorean theorem, if a right triangle has a hypotenuse with length <math alttext=\"c\"><mi>c</mi>\n</math> and legs with lengths <math alttext=\"a\"><mi>a</mi>\n</math> and <math alttext=\"b\"><mi>b</mi>\n</math>, then <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>. Substituting&nbsp;<math alttext=\"3 StartRoot 17 EndRoot\"><mn>3</mn><msqrt><mn>17</mn></msqrt></math> for <math alttext=\"c\"><mi>c</mi>\n</math> and <math alttext=\"3\"><mn>3</mn>\n</math> for <math alttext=\"b\"><mi>b</mi>\n</math> in this equation yields&nbsp;<math alttext=\"a squared plus left parenthesis 3 right parenthesis squared equals left parenthesis 3 StartRoot 17 EndRoot right parenthesis squared\"><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mfenced><mn>3</mn></mfenced><mn>2</mn></msup><mo>=</mo><msup><mfenced><mrow><mn>3</mn><msqrt><mn>17</mn></msqrt></mrow></mfenced><mn>2</mn></msup></math>, or&nbsp;<math alttext=\"a squared plus 9 equals 153\"><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mn>9</mn><mo>=</mo><mn>153</mn></math>. Subtracting <math alttext=\"9\"><mn>9</mn>\n</math> from both sides of this equation yields&nbsp;<math alttext=\"a squared equals 144\"><msup><mi>a</mi><mn>2</mn></msup><mo>=</mo><mn>144</mn></math>. Taking the square root of both sides of this equation yields&nbsp;<math alttext=\"a equals plus or minus StartRoot 144 EndRoot\"><mi>a</mi><mo>=</mo><mo>±</mo><msqrt><mn>144</mn></msqrt></math>, or&nbsp;<math alttext=\"a equals plus or minus 12\"><mi>a</mi><mo>=</mo><mo>±</mo><mn>12</mn></math>. Since <math alttext=\"a\"><mi>a</mi>\n</math> represents a length, which must be positive, the value of <math alttext=\"a\"><mi>a</mi>\n</math> is <math alttext=\"12\"><mn>12</mn>\n</math>. Thus, the length of the rectangle's longer side is <math alttext=\"12\"><mn>12</mn>\n</math>.</p>","correct_answer":["12"],"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}}