1
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Question

# Assume that Ai(i=1,2,.....,n) are the vertices of a regular polygon inscribed in a circle of radius unity. Find the value of |A1A2|2+|A1A3|2+.....+|A1An|2.

A
n
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B
n1
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C
4n
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D
2n
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Solution

## The correct option is D 2nGiven that Ai(i=1,2,......,n) are regular vertices of regular polygon |A1A2|2+|A1A3|2+.....+|A1An|2For a polygon, A1A2=2Rsinα.⇒A2A3=2Rsin(2α) and A1A4=2Rsin(3α)⇒A1An=2Rsin(nα)Where α=πn⇒12Rsinπ=12Rsin2π+12Rsin3π⇒1sinα=1sin2α+1sin3α⇒1sin2α=1sinα−1sin3α⇒1sin2α=sin3α−sinαsinα.sin3α⇒1sin2α=2sinα×cos2αsinα.sin3α⇒sin3α=2cos(2α)sin(2α)⇒sin3α=sin(4α)⇒3α=4α⇒3α−4α=π∴−α=π∴π+α=0∴|A1A2|2+|A1A3|2+.....+|A1An|2=2nHence, the ansswer is 2n.

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