{"id":"cb_question_bank:4e0ef4df-94fb-41e7-b8e9-df0d85a6f1ee","source_id":"cb_question_bank","source_item_id":"4e0ef4df-94fb-41e7-b8e9-df0d85a6f1ee","external_id":"1dd30616-5d7f-4401-8dd7-811d477d90d8","ibn":null,"question_id":"a4bd60a3","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 perimeter of an equilateral triangle is 624 centimeters. The height of this triangle is k StartRoot 3 EndRoot centimeters, where k is a constant. What is the value of k ?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: left;\">The perimeter of an equilateral triangle is <math alttext=\"624\"><mn>624</mn>\n</math> centimeters. The height of this triangle is <math alttext=\"k StartRoot 3 EndRoot\"><mi>k</mi><msqrt><mn>3</mn></msqrt></math> centimeters, where <math alttext=\"k\"><mi>k</mi>\n</math> is a constant. What is the value of <math alttext=\"k\"><mi>k</mi>\n</math>?</p>","rationale_html":"<p style=\"text-align: left;\">The correct answer is <math alttext=\"104\"><mn>104</mn>\n</math>. An equilateral triangle is a triangle in which all three sides have the same length and all three angles have a measure of <math alttext=\"60 degree\"><mn>60</mn><mo>°</mo></math>. The height of the triangle,&nbsp;<math alttext=\"k StartRoot 3 EndRoot\"><mi>k</mi><msqrt><mn>3</mn></msqrt></math>, is the length of the altitude from one vertex. The altitude divides the equilateral triangle into two congruent 30-60-90&nbsp;right triangles, where the altitude is the side across from the&nbsp;<math alttext=\"60 degree\"><mn>60</mn><mo>°</mo></math> angle in each&nbsp;30-60-90&nbsp;right triangle. Since the altitude has a length of <math alttext=\"k StartRoot 3 EndRoot\"><mi>k</mi><msqrt><mn>3</mn></msqrt></math>, it follows from the properties of&nbsp;30-60-90&nbsp;right triangles that the side across from each <math alttext=\"30 degree\"><mn>30</mn><mo>°</mo></math> angle has a length of <math alttext=\"k\"><mi>k</mi>\n</math> and each hypotenuse has a length of <math alttext=\"2 k\"><mrow>\n\t<mn>2</mn>\n\t<mi>k</mi>\n</mrow>\n</math>. In this case, the hypotenuse of each&nbsp;30-60-90&nbsp;right triangle is a side of the equilateral triangle; therefore, each side length of the equilateral triangle is <math alttext=\"2 k\"><mrow>\n\t<mn>2</mn>\n\t<mi>k</mi>\n</mrow>\n</math>. The perimeter of a triangle is the sum of the lengths of each side. It's given that the perimeter of the equilateral triangle is <math alttext=\"624\"><mn>624</mn>\n</math>; therefore, <math alttext=\"2 k plus 2 k plus 2 k equals 624\"><mn>2</mn><mi>k</mi><mo>+</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>2</mn><mi>k</mi><mo>=</mo><mn>624</mn></math>, or <math alttext=\"6 k equals 624\"><mn>6</mn><mi>k</mi><mo>=</mo><mn>624</mn></math>. Dividing both sides of this equation by <math alttext=\"6\"><mn>6</mn>\n</math> yields <math alttext=\"k equals 104\"><mrow>\n\t<mi>k</mi>\n\t<mo>=</mo>\n\t<mn>104</mn>\n</mrow>\n</math>.</p>","correct_answer":["104"],"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}}