Question

# Let A1,A2,....An be the vertices of an n−sided regular polygon such that 1A1A2=1A1A3+1A1A4. Then the value of n is:

A
7
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B
8
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C
9
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D
10
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Solution

## The correct option is A 7Let a be the side of n sided regular polygon A1,A2,A3,A4....An ∴ Angle subtended by each side at centre =2πn Also OA1=OA2=OA3=.....=OAn=r(Say) In ΔOA1A2, A1A2=2rsinπn Similarly in A1A3=2rsin2πn and in A1A4=2rsin3πn But given that 1A1A2=1A1A3+1A1A4 ⇒12sinπn=12sin2πn+12sin3πn ⇒1sinπn=sin3πn+sin2πnsin2πnsin3πn ⇒1sinπn=sin3πn+sin2πn2sinπncosπnsin3πn ⇒2cosπnsin3πn=sin3πn+sin2πn ⇒sin4πn+sin2πn=sin3πn+sin2πn ∴sin4πn=sin3πn ⇒sin4πn=sin(π−3πn) ⇒4πn=π−3πn ⇒7πn=π Thus, n=7

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