{"id":"cb_question_bank:a52c7729-5c50-4b44-a343-6f941232a4b8","source_id":"cb_question_bank","source_item_id":"a52c7729-5c50-4b44-a343-6f941232a4b8","external_id":"1597b05c-5fdb-4f34-a5b2-e92fa92dc557","ibn":null,"question_id":"ca2235f6","program":"SAT","module":"math","domain_code":"S","domain":"Geometry and Trigonometry","skill_code":"S.D.","skill":"Circles","difficulty":"H","score_band":6,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"A circle has center O , and points A and B lie on the circle. The measure of arc A B is 45 ∘ and the length of arc A B is 3 inches. What is the circumference, in inches, of the circle?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: left;\">A circle has center <math><mi>O</mi>\n</math>, and points <math><mi>A</mi>\n</math> and <math><mi>B</mi>\n</math> lie on the circle. The measure of arc <math><mrow>\n\t<mi>A</mi>\n\t<mi>B</mi>\n</mrow>\n</math> is <math><mrow><msup><mrow><mn>45</mn></mrow><mrow><mo>∘</mo></mrow></msup></mrow></math> and the length of arc <math><mrow>\n\t<mi>A</mi>\n\t<mi>B</mi>\n</mrow>\n</math> is <math><mn>3</mn>\n</math> inches. What is the circumference, in inches, of the circle?</p>","rationale_html":"<p>Choice D is correct. It’s given that the measure of arc <math alttext=\"upper A upper B\"><mi>A</mi><mi>B</mi></math> is&nbsp;<math alttext=\"45 degree\"><mn>45</mn><mo>°</mo></math> and the length of arc <math alttext=\"upper A upper B\"><mi>A</mi><mi>B</mi></math> is <math alttext=\"3 inches\"><mn>3</mn><mo>&nbsp;</mo><mtext>inches</mtext></math>. The arc measure of the full circle is <math alttext=\"360 degree\"><mn>360</mn><mo>°</mo></math>. If <math alttext=\"x\"><mi>x</mi>\n</math> represents the circumference, in inches, of the circle, it follows that <math alttext=\"StartFraction 45 degree Over 360 degree EndFraction equals StartFraction 3 inches Over x inches EndFraction\"><mfrac><mrow><mn>45</mn><mo>°</mo></mrow><mrow><mn>360</mn><mo>°</mo></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>3</mn><mo>&nbsp;</mo><mtext>inches</mtext></mrow><mrow><mi>x</mi><mo>&nbsp;</mo><mtext>inches</mtext></mrow></mfrac></math>. This equation is equivalent to <math alttext=\"StartFraction 45 Over 360 EndFraction equals StartFraction 3 Over x EndFraction\"><mfrac><mn>45</mn><mn>360</mn></mfrac><mo>=</mo><mfrac><mn>3</mn><mi>x</mi></mfrac></math>, or <math alttext=\"one eighth equals StartFraction 3 Over x EndFraction\"><mfrac><mn>1</mn><mn>8</mn></mfrac><mo>=</mo><mfrac><mn>3</mn><mi>x</mi></mfrac></math>. Multiplying both sides of this equation by <math alttext=\"8 x\"><mn>8</mn><mi>x</mi></math>&nbsp;yields <math alttext=\"1 left parenthesis x right parenthesis equals 3 left parenthesis 8 right parenthesis\"><mn>1</mn><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>3</mn><mo>(</mo><mn>8</mn><mo>)</mo></math>, or <math alttext=\"x equals 24\"><mi>x</mi><mo>=</mo><mn>24</mn></math>. Therefore, the circumference of the circle is <math alttext=\"24 inches\"><mn>24</mn><mo>&nbsp;</mo><mtext>inches</mtext></math>.</p>\n<p>Choice A is incorrect. This is the length of arc <math alttext=\"upper A upper B\"><mi>A</mi><mi>B</mi></math>.</p>\n<p>Choice B is incorrect and may result from multiplying the length of arc <math alttext=\"upper A upper B\"><mi>A</mi><mi>B</mi></math> by <math alttext=\"2\"><mn>2</mn>\n</math>.</p>\n<p>Choice C is incorrect and may result from squaring the length of arc <math alttext=\"upper A upper B\"><mi>A</mi><mi>B</mi></math>.</p>","correct_answer":["D"],"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":"<p><math><mn>3</mn>\n</math></p>","ord":0},{"label":"B","content_html":"<p><math><mn>6</mn>\n</math></p>","ord":1},{"label":"C","content_html":"<p><math><mn>9</mn>\n</math></p>","ord":2},{"label":"D","content_html":"<p><math><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}}