{"id":"cb_question_bank:f4faa03d-f945-42dc-9431-8c867b3862a1","source_id":"cb_question_bank","source_item_id":"f4faa03d-f945-42dc-9431-8c867b3862a1","external_id":"9657587b-66c2-4743-91d3-22e3bf8ab94e","ibn":null,"question_id":"459dd6c5","program":"SAT","module":"math","domain_code":"S","domain":"Geometry and Trigonometry","skill_code":"S.A.","skill":"Area and volume","difficulty":"H","score_band":7,"answer_type":"spr","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"Triangles italic upper A italic upper B italic upper C and italic upper D italic upper E italic upper F are similar. Each side length of triangle italic upper A italic upper B italic upper C is 4 times the corresponding side length of triangle italic upper D italic upper E italic upper F . The area of triangle italic upper A italic upper B italic upper C is 270 square inches. What is the area, in square inches, of triangle italic upper D italic upper E italic upper F ?","stimulus":"","stimulus_html":null,"stem_html":"<p>Triangles <em><math alttext=\"italic upper A italic upper B italic upper C\"><mtext>A</mtext><mtext>B</mtext><mtext>C</mtext></math></em> and <em><math alttext=\"italic upper D italic upper E italic upper F\"><mtext>D</mtext><mtext>E</mtext><mtext>F</mtext></math></em>&nbsp;are similar. Each side length of triangle <em><math alttext=\"italic upper A italic upper B italic upper C\"><mtext>A</mtext><mtext>B</mtext><mtext>C</mtext></math></em> is <math alttext=\"4\"><mn>4</mn>\n</math> times the corresponding side length of triangle <em><math alttext=\"italic upper D italic upper E italic upper F\"><mtext>D</mtext><mtext>E</mtext><mtext>F</mtext></math></em>. The area of triangle <em><math alttext=\"italic upper A italic upper B italic upper C\"><mtext>A</mtext><mtext>B</mtext><mtext>C</mtext></math></em> is <math alttext=\"270\"><mn>270</mn>\n</math> square inches. What is the area, in square inches, of triangle&nbsp;<math alttext=\"italic upper D italic upper E italic upper F\"><mtext>D</mtext><mtext>E</mtext><mtext>F</mtext></math>?</p>","rationale_html":"<p style=\"text-align: left;\">The correct answer is <math alttext=\"StartFraction 135 Over 8 EndFraction\"><mfrac>\n\t<mn>135</mn>\n\t<mn>8</mn>\n</mfrac>\n</math>. It's given that triangles <math alttext=\"italic upper A italic upper B italic upper C\"><mtext>A</mtext><mtext>B</mtext><mtext>C</mtext></math> and <math alttext=\"italic upper D italic upper E italic upper F\"><mtext>D</mtext><mtext>E</mtext><mtext>F</mtext></math> are similar and each side length of triangle <math alttext=\"italic upper A italic upper B italic upper C\"><mtext>A</mtext><mtext>B</mtext><mtext>C</mtext></math> is <math alttext=\"4\"><mn>4</mn>\n</math> times the corresponding side length of triangle <math alttext=\"italic upper D italic upper E italic upper F\"><mtext>D</mtext><mtext>E</mtext><mtext>F</mtext></math>. For two similar triangles, if each side length of the first triangle is <math alttext=\"k\"><mi>k</mi>\n</math> times the corresponding side length of the second triangle, then the area of the first triangle is <math alttext=\"k squared\"><msup>\n\t<mi>k</mi>\n\t<mn>2</mn>\n</msup>\n</math> times the area of the second triangle. Therefore, the area of triangle <math alttext=\"italic upper A italic upper B italic upper C\"><mtext>A</mtext><mtext>B</mtext><mtext>C</mtext></math> is <math alttext=\"4 squared\"><msup><mn>4</mn><mn>2</mn></msup></math>, or <math alttext=\"16\"><mn>16</mn>\n</math>, times the area of triangle <math alttext=\"italic upper D italic upper E italic upper F\"><mtext>D</mtext><mtext>E</mtext><mtext>F</mtext></math>. It's given that the area of triangle <math alttext=\"italic upper A italic upper B italic upper C\"><mtext>A</mtext><mtext>B</mtext><mtext>C</mtext></math> is <math alttext=\"270\"><mn>270</mn>\n</math> square inches. Let <math alttext=\"a\"><mi>a</mi>\n</math> represent the area, in square inches, of triangle <math alttext=\"italic upper D italic upper E italic upper F\"><mtext>D</mtext><mtext>E</mtext><mtext>F</mtext></math>. It follows that <math alttext=\"270\"><mn>270</mn>\n</math> is <math alttext=\"16\"><mn>16</mn>\n</math> times <math alttext=\"a\"><mi>a</mi>\n</math>, or <math alttext=\"270 equals 16 a\"><mrow>\n\t<mn>270</mn>\n\t<mo>=</mo>\n\t<mrow>\n\t\t<mn>16</mn>\n\t\t<mi>a</mi>\n\t</mrow>\n</mrow>\n</math>. Dividing both sides of this equation by <math alttext=\"16\"><mn>16</mn>\n</math> yields <math alttext=\"StartFraction 270 Over 16 EndFraction equals a\"><mfrac><mn>270</mn><mn>16</mn></mfrac><mo>=</mo><mi>a</mi></math>, which is equivalent to&nbsp;<math alttext=\"StartFraction 135 Over 8 EndFraction equals a\"><mfrac><mn>135</mn><mn>8</mn></mfrac><mo>=</mo><mi>a</mi></math>. Thus, the area, in square inches, of triangle <math alttext=\"italic upper D italic upper E italic upper F\"><mtext>D</mtext><mtext>E</mtext><mtext>F</mtext></math> is <math alttext=\"StartFraction 135 Over 8 EndFraction\"><mfrac>\n\t<mn>135</mn>\n\t<mn>8</mn>\n</mfrac>\n</math>. Note that 135/8, 16.87, and 16.88 are examples of ways to enter a correct answer.</p>","correct_answer":["135/8","16.87","16.88"],"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":[],"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}}