{"id":"cb_question_bank:8c39fe52-5b23-4791-8465-eb2d6a7a9293","source_id":"cb_question_bank","source_item_id":"8c39fe52-5b23-4791-8465-eb2d6a7a9293","external_id":"ae6da77a-4de7-4375-a532-b00350663a9f","ibn":null,"question_id":"f811d345","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":"A right triangle has legs with lengths of 24 centimeters and 21 centimeters. If the length of this triangle's hypotenuse, in centimeters, can be written in the form 3 StartRoot d EndRoot , where d is an integer, what is the value of d ?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: left;\">A right triangle has legs with lengths of <math alttext=\"24\"><mn>24</mn>\n</math> centimeters and <math alttext=\"21\"><mn>21</mn>\n</math> centimeters. If the length of this triangle's hypotenuse, in centimeters, can be written in the form <math alttext=\"3 StartRoot d EndRoot\"><mrow><mn>3</mn></mrow><msqrt><mi>d</mi></msqrt></math>, where <math alttext=\"d\"><mi>d</mi>\n</math> is an integer, what is the value of <math alttext=\"d\"><mi>d</mi>\n</math>?</p>","rationale_html":"<p style=\"text-align: left;\">The correct answer is <math alttext=\"113\"><mn>113</mn>\n</math>. It's given that the legs of a right triangle have lengths <math alttext=\"24\"><mn>24</mn>\n</math> centimeters and <math alttext=\"21\"><mn>21</mn>\n</math> centimeters. 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 two legs. It follows that if <math alttext=\"h\"><mi>h</mi>\n</math> represents the length, in centimeters, of the hypotenuse of the right triangle, <math alttext=\"h squared equals 24 squared plus 21 squared\"><msup><mi>h</mi><mn>2</mn></msup><mo>=</mo><msup><mn>24</mn><mn>2</mn></msup><mo>+</mo><msup><mn>21</mn><mn>2</mn></msup></math>. This equation is equivalent to <math alttext=\"h squared equals 1,017\"><mrow>\n\t<msup>\n\t\t<mi>h</mi>\n\t\t<mn>2</mn>\n\t</msup>\n\t<mo>=</mo>\n\t<mn>1,017</mn>\n</mrow>\n</math>. Taking the square root of each side of this equation yields <math alttext=\"h equals StartRoot 1,017 EndRoot\"><mi>h</mi><mo>=</mo><msqrt><mn>1,017</mn></msqrt></math>. This equation can be rewritten as <math alttext=\"h equals StartRoot 9 dot 113 EndRoot\"><mi>h</mi><mo>=</mo><msqrt><mn>9</mn><mo>·</mo><mn>113</mn></msqrt></math>, or&nbsp;<math alttext=\"h equals StartRoot 9 EndRoot dot StartRoot 113 EndRoot\"><mi>h</mi><mo>=</mo><msqrt><mn>9</mn></msqrt><mo>·</mo><msqrt><mn>113</mn></msqrt></math>. This equation is equivalent to <math alttext=\"h equals 3 StartRoot 113 EndRoot\"><mrow>\n\t<mi>h</mi>\n\t<mo>=</mo>\n\t<mrow>\n\t\t<mn>3</mn>\n\t\t<msqrt>\n\t\t\t<mn>113</mn>\n\t\t</msqrt>\n\t</mrow>\n</mrow>\n</math>. It's given that the length of the triangle's hypotenuse, in centimeters, can be written in the form <math alttext=\"3 StartRoot d EndRoot\"><mrow>\n\t<mn>3</mn>\n\t<msqrt>\n\t\t<mi>d</mi>\n\t</msqrt>\n</mrow>\n</math>. It follows that the value of <math alttext=\"d\"><mi>d</mi>\n</math> is <math alttext=\"113\"><mn>113</mn>\n</math>.</p>","correct_answer":["113"],"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}}