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Question

If A1,A2........An are the vertices of a regular plane polygon with n sides and O is the center then n1i=1(OAi×OAi+1)=

A
(n+1)(OA1×OA2)
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B
(n+1)(OA2×OA1)
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C
(n1) (OA1×OA2)
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D
(n1) (OA2×OA1)
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Solution

The correct option is B (n1) (OA1×OA2)
n1i=1(¯¯¯¯¯¯¯¯¯¯OAiׯ¯¯¯¯¯¯¯¯¯¯¯¯¯¯OAi+1)
¯¯¯¯¯¯¯¯¯¯OA1ׯ¯¯¯¯¯¯¯¯¯OA2=|¯¯¯¯¯¯¯¯¯¯OA1||¯¯¯¯¯¯¯¯¯¯OA2|sinθ^n
¯¯¯¯¯¯¯¯¯¯OA1,¯¯¯¯¯¯¯¯¯¯OA2,......¯¯¯¯¯¯¯¯¯¯¯OAn will be of same length and subtend same angle at the centre.
n1i=1(¯¯¯¯¯¯¯¯¯¯OAiׯ¯¯¯¯¯¯¯¯¯¯¯¯¯¯OAi+1)=(n1)|¯¯¯¯¯¯¯¯¯¯OA1||¯¯¯¯¯¯¯¯¯¯OA2|sinθ^n
n1i=1(¯¯¯¯¯¯¯¯¯¯OAiׯ¯¯¯¯¯¯¯¯¯¯¯¯¯¯OAi+1)=(n1)(¯¯¯¯¯¯¯¯¯¯OA1ׯ¯¯¯¯¯¯¯¯¯OA2)
Hence, option C is correct.

652041_37229_ans_46ab08686f794ee096fc4490e997b02a.png

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