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Elemente Der Mathematik Aufgaben: Christopher Zhang

The document discusses the locus of centroids of a sector of a circle as the central angle varies from 0 to 2π radians. It finds that the centroid is the origin when the central angle is π radians.

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50 views1 page

Elemente Der Mathematik Aufgaben: Christopher Zhang

The document discusses the locus of centroids of a sector of a circle as the central angle varies from 0 to 2π radians. It finds that the centroid is the origin when the central angle is π radians.

Uploaded by

Duo Head
Copyright
© Attribution Non-Commercial (BY-NC)
We take content rights seriously. If you suspect this is your content, claim it here.
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Elemente der Mathematik Aufgaben

Christopher Zhang
1227. A sector of a circle has radius r and central angle . Its vertex is the origin and has a leg on the positive x-axis. Find the locus of centroids as varies in the interval (0, 2 ). For what 0 is the centroid the origin? Solution: Let O be the origin and A be (r, 0). Let B = O be the endpoint of the other leg of the sector. Let G be the centroid of OAB and D the O median. For (0, ], AD = r cos . 2 2 2r 2r AG = AD = cos . Thus on (0, ], the locus of is dened by the function in polar R = cos . 3 3 2 3 2 2r 2 2r For (, 2 ), AOD = 2 , thus AG = cos = cos . Thus the locus of points G 3 2 3 2 is dened by the polar function 2r cos (0, ] f () = 3 2r 2 3 cos 2 (, 2 ] G = O if and only if R = 0 or when = .

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