{"id":"cb_question_bank:eef88f14-05ab-49cc-a68b-d45c42c8de02","source_id":"cb_question_bank","source_item_id":"eef88f14-05ab-49cc-a68b-d45c42c8de02","external_id":"95dd10f0-9b55-4202-a756-91f1ede906ac","ibn":null,"question_id":"306264ab","program":"SAT","module":"math","domain_code":"S","domain":"Geometry and Trigonometry","skill_code":"S.A.","skill":"Area and volume","difficulty":"H","score_band":6,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"A right triangle has sides of length 2 StartRoot 2 EndRoot , 6 StartRoot 2 EndRoot , and StartRoot 80 EndRoot units. What is the area of the triangle, in square units?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: left;\">A right triangle has sides of length <math alttext=\"2 StartRoot 2 EndRoot\"><mrow>\n\t<mn>2</mn>\n\t<msqrt>\n\t\t<mn>2</mn>\n\t</msqrt>\n</mrow>\n</math>, <math alttext=\"6 StartRoot 2 EndRoot\"><mrow>\n\t<mn>6</mn>\n\t<msqrt>\n\t\t<mn>2</mn>\n\t</msqrt>\n</mrow>\n</math>, and&nbsp;<math alttext=\"StartRoot 80 EndRoot\"><msqrt><mrow><mn>80</mn></mrow></msqrt></math> units. What is the area of the triangle, in square units?</p>","rationale_html":"<p>Choice B is correct. The area, <math alttext=\"upper A\"><mi>A</mi>\n</math>, of a triangle can be found using the formula <math alttext=\"upper A equals one half b h\"><mi>A</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>b</mi><mi>h</mi></math>, where <math alttext=\"b\"><mi>b</mi>\n</math> is the length of the base of the triangle and <math alttext=\"h\"><mi>h</mi>\n</math> is the height of the triangle. It's given that the triangle is a right triangle. Therefore, its base and height can be represented by the two legs. It’s also given that the triangle has sides of length <math alttext=\"2 StartRoot 2 EndRoot\"><mn>2</mn><msqrt><mn>2</mn></msqrt></math>, <math alttext=\"6 StartRoot 2 EndRoot\"><mn>6</mn><msqrt><mn>2</mn></msqrt></math>, and <math alttext=\"StartRoot 80 EndRoot\"><msqrt><mn>80</mn></msqrt></math> units. Since&nbsp;<math alttext=\"StartRoot 80 EndRoot\"><msqrt><mn>80</mn></msqrt></math> units is the greatest of these lengths, it's the length of the hypotenuse. Therefore, the two legs have lengths <math alttext=\"2 StartRoot 2 EndRoot\"><mn>2</mn><msqrt><mn>2</mn></msqrt></math> and <math alttext=\"6 StartRoot 2 EndRoot\"><mn>6</mn><msqrt><mn>2</mn></msqrt></math> units. Substituting these values for <math alttext=\"b\"><mi>b</mi>\n</math> and <math alttext=\"h\"><mi>h</mi>\n</math> in the formula&nbsp;<math alttext=\"upper A equals one half b h\"><mi>A</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>b</mi><mi>h</mi></math> gives <math alttext=\"upper A equals one half left parenthesis 2 StartRoot 2 EndRoot right parenthesis left parenthesis 6 StartRoot 2 EndRoot right parenthesis\"><mi>A</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced><mrow><mn>2</mn><msqrt><mn>2</mn></msqrt></mrow></mfenced><mfenced><mrow><mn>6</mn><msqrt><mn>2</mn></msqrt></mrow></mfenced></math>, which is equivalent to <math alttext=\"upper A equals 6 StartRoot 4 EndRoot\"><mi>A</mi><mo>=</mo><mn>6</mn><msqrt><mn>4</mn></msqrt></math> square units, or <math alttext=\"upper A equals 12\"><mrow>\n\t<mi>A</mi>\n\t<mo>=</mo>\n\t<mn>12</mn>\n</mrow>\n</math> square units.</p>\n<p>Choice A is incorrect. This expression represents the perimeter, rather than the area, of the triangle.</p>\n<p>Choice C is incorrect and may result from conceptual or calculation errors.&nbsp;</p>\n<p>Choice D is incorrect and may result from conceptual or calculation errors.&nbsp;</p>","correct_answer":["B"],"has_media":false,"has_mathml":true,"has_table":false,"source_created_at":"2023-08-02T20:25:59.837000+00:00","source_updated_at":"2023-08-02T20:25:59.837000+00:00","options":[{"label":"A","content_html":"<p><math alttext=\"8 StartRoot 2 EndRoot plus StartRoot 80 EndRoot\"><mrow>\n\t<mn>8</mn>\n\t<msqrt>\n\t\t<mn>2</mn>\n\t</msqrt>\n</mrow>\n<mo>+</mo><msqrt><mrow><mn>80</mn></mrow></msqrt></math></p>","ord":0},{"label":"B","content_html":"<p><math alttext=\"12\"><mn>12</mn>\n</math></p>","ord":1},{"label":"C","content_html":"<p><math alttext=\"24 StartRoot 80 EndRoot\"><mn>24</mn>\n<msqrt><mrow><mn>80</mn></mrow></msqrt></math></p>","ord":2},{"label":"D","content_html":"<p><math alttext=\"24\"><mn>24</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}}