{"id":"cb_question_bank:71620d76-9268-47de-85bf-63b4034e053b","source_id":"cb_question_bank","source_item_id":"71620d76-9268-47de-85bf-63b4034e053b","external_id":"437a1594-67bd-4051-8026-2a747bd29288","ibn":null,"question_id":"24cec8d1","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 has center upper O , and points upper R and upper S lie on the circle. In triangle upper O upper R upper S , the measure of angle upper R upper O upper S is 88 degree . What is the measure of angle upper R upper S upper O , in degrees? (Disregard the degree symbol when entering your answer.)","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: left;\">A circle has center <math alttext=\"upper O\"><mi>O</mi>\n</math>, and points <math alttext=\"upper R\"><mi>R</mi>\n</math> and <math alttext=\"upper S\"><mi>S</mi>\n</math> lie on the circle. In triangle <math alttext=\"upper O upper R upper S\"><mrow>\n\t<mi>O</mi>\n\t<mi>R</mi>\n\t<mi>S</mi>\n</mrow>\n</math>, the measure of <math alttext=\"angle upper R upper O upper S\"><mo>∠</mo><mi>R</mi><mi>O</mi><mi>S</mi></math> is <math alttext=\"88 degree\"><mrow><mn>88</mn></mrow><mo>°</mo></math>. What is the measure of <math alttext=\"angle upper R upper S upper O\"><mo>∠</mo><mi>R</mi><mi>S</mi><mi>O</mi></math>, in degrees? (Disregard the degree symbol when entering your answer.)</p>","rationale_html":"<p style=\"text-align: left;\">The correct answer is <math alttext=\"46\"><mn>46</mn>\n</math>. It's given that <math alttext=\"upper O\"><mi>O</mi>\n</math> is the center of a circle and that points <math alttext=\"upper R\"><mi>R</mi>\n</math> and <math alttext=\"upper S\"><mi>S</mi>\n</math> lie on the circle. Therefore,&nbsp;<math alttext=\"ModifyingAbove upper O upper R With bar\"><mover><mrow><mi>O</mi><mi>R</mi></mrow><mo>¯</mo></mover></math> and&nbsp;<math alttext=\"ModifyingAbove upper O upper S With bar\"><mover><mrow><mi>O</mi><mi>S</mi></mrow><mo>¯</mo></mover></math> are radii of the circle. It follows that&nbsp;<math alttext=\"upper O upper R equals upper O upper S\"><mi>O</mi><mi>R</mi><mo>=</mo><mi>O</mi><mi>S</mi></math>. If two sides of a triangle are congruent, then the angles opposite them are congruent. It follows that the angles <math alttext=\"angle upper R upper S upper O\"><mo>∠</mo><mi>R</mi><mi>S</mi><mi>O</mi></math> and <math alttext=\"angle upper O upper R upper S\"><mo>∠</mo><mi>O</mi><mi>R</mi><mi>S</mi></math>, which are across from the sides of equal length, are congruent. Let&nbsp;<math alttext=\"x degree\"><mi>x</mi><mo>°</mo></math> represent the measure of <math alttext=\"angle upper R upper S upper O\"><mo>∠</mo><mi>R</mi><mi>S</mi><mi>O</mi></math>. It follows that the measure of&nbsp;<math alttext=\"angle upper O upper R upper S\"><mo>∠</mo><mi>O</mi><mi>R</mi><mi>S</mi></math> is also <math alttext=\"x degree\"><mi>x</mi><mo>°</mo></math>. It's given that the measure of&nbsp;<math alttext=\"angle upper R upper O upper S\"><mo>∠</mo><mi>R</mi><mi>O</mi><mi>S</mi></math> is <math alttext=\"88 degree\"><mn>88</mn><mo>°</mo></math>. Because the sum of the measures of the interior angles of a triangle is <math alttext=\"180 degree\"><mn>180</mn><mo>°</mo></math>, the equation <math alttext=\"x degree plus x degree plus 88 degree  equals 180 degree\"><mi>x</mi><mo>°</mo><mo>+</mo><mi>x</mi><mo>°</mo><mo>+</mo><mn>88</mn><mo>°</mo><mo>=</mo><mn>180</mn><mo>°</mo></math>, or&nbsp;<math alttext=\"2 x plus 88 equals 180\"><mn>2</mn><mi>x</mi><mo>+</mo><mn>88</mn><mo>=</mo><mn>180</mn></math>, can be used to find the measure of <math alttext=\"angle upper R upper S upper O\"><mo>∠</mo><mi>R</mi><mi>S</mi><mi>O</mi></math>. Subtracting <math alttext=\"88\"><mn>88</mn>\n</math>&nbsp;from both sides of this equation yields&nbsp;<math alttext=\"2 x equals 92\"><mn>2</mn><mi>x</mi><mo>=</mo><mn>92</mn></math>. Dividing both sides of this equation by <math alttext=\"2\"><mn>2</mn>\n</math> yields&nbsp;<math alttext=\"x equals 46\"><mi>x</mi><mo>=</mo><mn>46</mn></math>. Therefore, the measure of <math alttext=\"angle upper R upper S upper O\"><mo>∠</mo><mi>R</mi><mi>S</mi><mi>O</mi></math>, in degrees, is <math alttext=\"46\"><mn>46</mn>\n</math>.</p>","correct_answer":["46"],"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}}