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

The circle below has center $O$, the length of arc $\\overset{\\large\\frown}{ADC}$ is $5\\pi$, and $x=100$. What is the length of the arc $\\overset{\\large\\frown}{ABC}$

$\"\"$

","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-70"}],"about":{"@type":"Thing","name":"Math, Geometry and Trigonometry, Duplicate"},"educationalLevel":"intermediate","eduQuestionType":"Multiple choice","learningResourceType":"Practice problem","assesses":["Math","Geometry and Trigonometry","Duplicate"],"name":"

The circle below has center $O$, the length of arc $\\overset{\\large\\frown}{ADC}$ is $5\\pi$, and $x=100$. What is the length of the arc $\\overset{\\large\\frown}{ABC}$

$\"\"$

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

Choice B is correct. The ratio of the lengths of two arcs of a circle is equal to the ratio of the measures of the central angles that subtend the arcs. It's given that arc $\\overset{\\large\\frown}{ADC}$ is subtended by a central angle with measure $100^{\\circ}$. Since the sum of the measures of the angles about a point is $360^{\\circ}$, it follows that arc $\\overset{\\large\\frown}{ABC}$ is subtended by a central angle with measure $360^{\\circ}-100^{\\circ}=260^{\\circ}$. If $s$ is the length of arc $\\overset{\\large\\frown}{ABC}$, then $s$ must satisfy the ratio $\\dfrac{s}{5\\pi}=\\dfrac{260}{100}$. Reducing the fraction $\\dfrac{260}{100}$ to its simplest form gives $\\dfrac{13}{5}$. Therefore, $\\dfrac{s}{5\\pi}=\\dfrac{13}{5}$. Multiplying both sides of $\\dfrac{s}{5\\pi}=\\dfrac{13}{5}$ by $5\\pi$ yields $s=13\\pi$.

Choice A is incorrect. This is the length of an arc consisting of exactly half of the circle, but arc $\\overset{\\large\\frown}{ABC}$ is greater than half of the circle. Choice C is incorrect. This is the total circumference of the circle. Choice D is incorrect. This is half the length of arc $\\overset{\\large\\frown}{ABC}$, not its full length.

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

The circle below has center $O$, the length of arc $\\overset{\\large\\frown}{ADC}$ is $5\\pi$, and $x=100$. What is the length of the arc $\\overset{\\large\\frown}{ABC}$

$\"\"$

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

$9\\pi$

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

$18\\pi$

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

$\\dfrac{13}{2}\\pi$

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

$13\\pi$

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

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

"}}},"url":"https://classnotes.xyz/mcqs/sat-math-geometry-and-trigonometry-multiple-choice-question-70"}

The circle below has center $O$, the length of arc $\overset{\large\frown}{ADC}$ is $5\pi$, and $x=100$. What is the length of the arc $\overset{\large\frown}{ABC}$