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| − | ==Problem ==
| + | #REDIRECT [[2018_AMC_10B_Problems/Problem_12]] |
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| − | Line segment <math>\overline{AB}</math> is a diameter of a circle with <math>AB = 24</math>. Point <math>C</math>, not equal to <math>A</math> or <math>B</math>, lies on the circle. As point <math>C</math> moves around the circle, the centroid (center of mass) of <math>\triangle ABC</math> traces out a closed curve missing two points. To the nearest positive integer, what is the area of the region bounded by this curve?
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| − | <math>\textbf{(A) } 25 \qquad \textbf{(B) } 38 \qquad \textbf{(C) } 50 \qquad \textbf{(D) } 63 \qquad \textbf{(E) } 75 </math>
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| − | ==Solution==
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| − | For each <math>\triangle ABC,</math> note that the length of one median is <math>OC=12.</math> Let <math>G</math> be the centroid of <math>\triangle ABC.</math> It follows that <math>OG=\frac13 OC=4.</math>
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| − | Two shapes of <math>\triangle ABC,</math> namely <math>\triangle ABC_1</math> and <math>\triangle ABC_2</math> with their respective centroids <math>G_1</math> and <math>G_2,</math> are shown below:
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| − | <b>DIAGRAM NEEDED</b>
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| − | Therefore, point <math>G</math> traces out a circle (missing two points) with the center <math>O</math> and the radius <math>\overline{OG},</math> as indicated in red. To the nearest positive integer, the area of the region bounded by the red curve is <math>\pi\cdot OG^2=16\pi\approx\boxed{\textbf{(C) } 50}.</math>
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| − | ~MRENTHUSIASM
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| − | ==See Also==
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| − | {{AMC12 box|year=2018|ab=B|num-a=9|num-b=7}}
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| − | {{MAA Notice}}
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| − | [[Category:Intermediate Geometry Problems]] | |
