{"id":"cb_question_bank:7150265e-6b41-431e-9bd3-c357faf60fd0","source_id":"cb_question_bank","source_item_id":"7150265e-6b41-431e-9bd3-c357faf60fd0","external_id":"4959ea4c-8faf-4a7c-9ef9-60dc6d45d751","ibn":null,"question_id":"ffe862a3","program":"SAT","module":"math","domain_code":"S","domain":"Geometry and Trigonometry","skill_code":"S.C.","skill":"Right triangles and trigonometry","difficulty":"H","score_band":7,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"An isosceles right triangle has a hypotenuse of length 58 inches. What is the perimeter, in inches, of this triangle?","stimulus":"","stimulus_html":null,"stem_html":"

An isosceles right triangle has a hypotenuse of length 58\n inches. What is the perimeter, in inches, of this triangle?

","rationale_html":"

Choice C is correct. Since the triangle is an isosceles right triangle, the two sides that form the right angle must be the same length. Let x\n be the length, in inches, of each of those sides. The Pythagorean theorem states that in a right triangle, \n\t\n\t\t\n\t\t\ta\n\t\t\t2\n\t\t\n\t\t+\n\t\t\n\t\t\tb\n\t\t\t2\n\t\t\n\t\n\t=\n\t\n\t\tc\n\t\t2\n\t\n\n, where c\n is the length of the hypotenuse and a\n and b\n are the lengths of the other two sides. Substituting x\n for a\n, x\n for b\n, and 58\n for c\n in this equation yields x2+x2=582, or 2x2=582. Dividing each side of this equation by 2\n yields x2=5822, or x2=2·5824. Taking the square root of each side of this equation yields two solutions: x=5822 and x=-5822. The value of x\n must be positive because it represents a side length. Therefore, x=5822, or \n\tx\n\t=\n\t\n\t\t29\n\t\t\n\t\t\t2\n\t\t\n\t\n\n. The perimeter, in inches, of the triangle is 58+x+x, or 58+2x. Substituting \n\t29\n\t\n\t\t2\n\t\n\n for x\n in this expression gives a perimeter, in inches, of 58+2292, or \n\t58\n\t+\n\t\n\t\t58\n\t\t\n\t\t\t2\n\t\t\n\t\n\n.

\n

Choice A is incorrect. This is the length, in inches, of each of the congruent sides of the triangle, not the perimeter, in inches, of the triangle.

\n

Choice B is incorrect. This is the sum of the lengths, in inches, of the congruent sides of the triangle, not the perimeter, in inches, of the triangle.

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Choice D is incorrect and may result from conceptual or calculation errors.

","correct_answer":["C"],"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":"

\n\t29\n\t\n\t\t2\n\t\n\n

","ord":0},{"label":"B","content_html":"

\n\t58\n\t\n\t\t2\n\t\n\n

","ord":1},{"label":"C","content_html":"

\n\t58\n\t+\n\t\n\t\t58\n\t\t\n\t\t\t2\n\t\t\n\t\n\n

","ord":2},{"label":"D","content_html":"

\n\t58\n\t+\n\t\n\t\t116\n\t\t\n\t\t\t2\n\t\t\n\t\n\n

","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}}