{"@context":"https://schema.org/","@type":"Quiz","name":"

Point $O$ is the center of the circle below, and the measure of $\\angle OAB$ is $30^{\\circ}$. If the length of $\\overline{OC}$ is 18, what is the length of arc $\\overset{\\large\\frown}{AB}$?

$\"\"$

","hasPart":{"@type":"Question","typicalAgeRange":"12-22","educationalAlignment":[{"@type":"AlignmentObject","alignmentType":"educationalSubject","targetName":"SAT"},{"@type":"AlignmentObject","alignmentType":"educationalLevel","targetName":"SAT","educationalFramework":"Common Core","targetUrl":"https://classnotes.xyz/mcqs/sat-math-geometry-and-trigonometry-multiple-choice-question-72"}],"about":{"@type":"Thing","name":"Math, Geometry and Trigonometry, Duplicate"},"educationalLevel":"intermediate","eduQuestionType":"Multiple choice","learningResourceType":"Practice problem","assesses":["Math","Geometry and Trigonometry","Duplicate"],"name":"

Point $O$ is the center of the circle below, and the measure of $\\angle OAB$ is $30^{\\circ}$. If the length of $\\overline{OC}$ is 18, what is the length of arc $\\overset{\\large\\frown}{AB}$?

$\"\"$

","comment":{"@type":"Comment","text":"

Choice B is correct. Because segments $OA$ and $OB$ are radii of the circle centered at point $O$, these segments have equal lengths. Therefore, triangle $AOB$ is an isosceles triangle, where angles $OAB$ and $OBA$ are congruent base angles of the triangle. It's given that angle $OAB$ measures $30^{\\circ}$. Therefore, angle OBA also measures $30^{\\circ}$. Let $x^{\\circ}$ represent the measure of angle $AOB$. Since the sum of the measures of the three angles of any triangle is $180^{\\circ}$, it follows that $30^{\\circ}+30^{\\circ}+x^{\\circ}=180^{\\circ}$, or $60^{\\circ}+x^{\\circ}=180^{\\circ}$. Subtracting $60^{\\circ}$ from both sides of this equation yields $x^{\\circ}=120^{\\circ}$, or $\\dfrac{2\\pi}{3}$ radians. Therefore, the measure of angle $AOB$, and thus the measure of arc $\\overset{\\large\\frown}{AB}$, is $\\dfrac{2\\pi}{3}$ radians. Since $\\overline{OC}$ is a radius of the given circle and its length is 18, the length of the radius of the circle is 18. Therefore, the length of arc $\\overset{\\large\\frown}{AB}$ can be calculated as $\\left(\\dfrac{2\\pi}{3}\\right)(18)$, or $12\\pi$.

Choices A, C, and D are incorrect and may result from conceptual or computational errors.

"},"encodingFormat":"text/markdown","text":"

Point $O$ is the center of the circle below, and the measure of $\\angle OAB$ is $30^{\\circ}$. If the length of $\\overline{OC}$ is 18, what is the length of arc $\\overset{\\large\\frown}{AB}$?

$\"\"$

","suggestedAnswer":[{"@type":"Answer","position":0,"encodingFormat":"text/html","text":"

$9\\pi$

"},{"@type":"Answer","position":2,"encodingFormat":"text/html","text":"

$15\\pi$

"},{"@type":"Answer","position":3,"encodingFormat":"text/html","text":"

$18\\pi$

"}],"acceptedAnswer":{"@type":"Answer","position":1,"encodingFormat":"text/html","text":"

$12\\pi$

","comment":{"@type":"Comment","text":"

Choices A, C, and D are incorrect and may result from conceptual or computational errors.

"},"answerExplanation":{"@type":"Comment","text":"

Choices A, C, and D are incorrect and may result from conceptual or computational errors.

"}}},"url":"https://classnotes.xyz/mcqs/sat-math-geometry-and-trigonometry-multiple-choice-question-72"}

Point $O$ is the center of the circle below, and the measure of $\angle OAB$ is $30^{\circ}$. If the length of $\overline{OC}$ is 18, what is the length of arc $\overset{\large\frown}{AB}$?