{"id":"cb_question_bank:3cabdafb-1ad5-45aa-b597-43ce6f0b0da8","source_id":"cb_question_bank","source_item_id":"3cabdafb-1ad5-45aa-b597-43ce6f0b0da8","external_id":"58f8a50f-ca96-4d35-9e5d-bb5694488742","ibn":null,"question_id":"ebbf23ae","program":"SAT","module":"math","domain_code":"S","domain":"Geometry and Trigonometry","skill_code":"S.D.","skill":"Circles","difficulty":"H","score_band":6,"answer_type":"spr","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"A circle in the xy -plane has a diameter with endpoints left parenthesis 2 comma 4 right parenthesis and left parenthesis 2 comma 14 right parenthesis . An equation of this circle is left parenthesis x minus 2 right parenthesis squared plus left parenthesis y minus 9 right parenthesis squared equals r squared , where r is a positive constant. What is the value of r ?","stimulus":"","stimulus_html":null,"stem_html":"

A circle in the xy-plane has a diameter with endpoints 2,4 and 2,14. An equation of this circle is \n\t\n\t\t\n\t\t\t\n\t\t\t\t\n\t\t\t\t\tx\n\t\t\t\t\t-\n\t\t\t\t\t2\n\t\t\t\t\n\t\t\t\n\t\t\t2\n\t\t\n\t\t+\n\t\t\n\t\t\t\n\t\t\t\t\n\t\t\t\t\ty\n\t\t\t\t\t-\n\t\t\t\t\t9\n\t\t\t\t\n\t\t\t\n\t\t\t2\n\t\t\n\t\n\t=\n\t\n\t\tr\n\t\t2\n\t\n\n, where r\n is a positive constant. What is the value of r\n?

","rationale_html":"

The correct answer is 5\n. The standard form of an equation of a circle in the xy-plane is x-h2+y-k2=r2, where h\n, k\n, and r\n are constants, the coordinates of the center of the circle are h,k, and the length of the radius of the circle is r\n. It′s given that an equation of the circle is x-22+y-92=r2. Therefore, the center of this circle is 2,9. It’s given that the endpoints of a diameter of the circle are 2,4 and 2,14. The length of the radius is the distance from the center of the circle to an endpoint of a diameter of the circle, which can be found using the distance formula, x1-x22+y1-y22. Substituting the center of the circle 2,9 and one endpoint of the diameter 2,4 in this formula gives a distance of 2-22+9-42, or 02+52, which is equivalent to 5\n. Since the distance from the center of the circle to an endpoint of a diameter is 5\n, the value of r\n is 5\n.

","correct_answer":["5"],"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}}