{"id":"cb_practice_pdf:t6-math-m2-q25","source_id":"cb_practice_pdf","source_item_id":"t6-math-m2-q25","external_id":null,"ibn":null,"question_id":null,"program":"SAT","module":"math","domain_code":null,"domain":null,"skill_code":null,"skill":null,"difficulty":null,"score_band":null,"answer_type":"mcq","answer_source":"rationale","layout_warning":"Part of this question was typeset outside the text flow — a fraction, subscript or similar — and PDF extraction re-inserted it in the wrong place. The wording is reliable; the symbols may be out of order.","form_id":"cb_practice_pdf:test-6","module_number":2,"position":25,"stem":"In the xy-plane, a circle has center C with (h,k) coordinates. Points A and B lie on the circle. J_ h +1,k+ 102 Point A has coordinates, and ACB AB is a right angle. What is the length of? J_ 206","stimulus":"","stimulus_html":null,"stem_html":"<p>In the xy-plane, a circle has center C with (h,k) coordinates. Points A and B lie on the circle. J_ h +1,k+ 102 Point A has coordinates, and ACB AB is a right angle. What is the length of? J_ 206</p>","rationale_html":"<p>Choice A is correct. It’s given that points A and B lie on the circle with center C. Therefore, AC and BC are both radii of the circle. Since all radii of a circle are congruent, AC is congruent to BC. The length of AC, or the distance from point A to point C, can be found using the distance formula, which gives the distance between two points, ^x1 , y1 h and ^x2 , y2 h, as ^x1- x2 h 2+ ^y1- y2 h 2 . Substituting the given coordinates of point A, ^h +1, k + 102 h, for ^x1 , y1 h and , the given coordinates of point C, ^h,kh, for ^x2 y2 h in the distance formula yields +1- 2+ + 102- 2 12+ 102 2 ^h hh ^k kh , or ^ h , which is equivalent to 1+102 103 103 , or . Therefore, the length of AC is and the length of BC is 103 . It’s given that angle ACB is a right angle. Therefore, triangle ACB is a right triangle with legs AC and BC and hypotenuse AB. By the Pythagorean theorem, if a right triangle has a hypotenuse with length c and legs with lengths a and b, 2+ 2= 2 103 then a b c . Substituting for a and b in this equation yields 103 2 + 103 2 = 2 103+103= 2 206= 2 ^ h ^ h c , or c , which is equivalent to c . Taking 206= the positive square root of both sides of this equation yields c. Therefore, 206 the length of AB is . Choice B is incorrect and may result from conceptual or calculation errors. Choice C is incorrect. This would be the length of AB if the length of AC were 103 103 , not . Choice D is incorrect and may result from conceptual or calculation errors.</p>","correct_answer":["A"],"has_media":false,"has_mathml":false,"has_table":false,"source_created_at":null,"source_updated_at":null,"options":[{"label":"A","content_html":"<p>2 J_ 102</p>","ord":0},{"label":"B","content_html":"<p>103 2 j</p>","ord":1},{"label":"C","content_html":"<p>103 3 j</p>","ord":2},{"label":"D","content_html":"","ord":3}],"attribution":{"source_id":"cb_practice_pdf","source_name":"College Board full-length SAT practice tests (PDF)","rights_holder":"College Board","canonical_url":"https://satsuite.collegeboard.org/practice/practice-tests/paper","retrieved_from":"https://satsuite.collegeboard.org/practice/practice-tests/paper","attribution":"Full-length SAT practice tests 4–11 © College Board, published free at satsuite.collegeboard.org.","disclaimer":"Extracted from PDF layout: equations, figures and tables do not survive intact. Use the original PDFs for anything the text renders poorly. SAT® is a trademark registered by the College Board, which is not affiliated with and does not endorse this project.","license":"College Board copyright. Free to download; not licensed for redistribution.","redistributable":false,"item_count":905}}