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Question

# If A1A2A3...An be a regular polygon of n sides and 1A1A2=1A1A3+1A1A4,then

A
n=5
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B
n=6
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C
n=7
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D
none of these.
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Solution

## The correct option is C n=7If radius of circle is r then A1A2=2rsin(πn)A1A3=2rsin(2πn)A1A4=2rsin(3πn)∵1A1A2=1A1A3+1A1A4⇒12rsin(πn)=12rsin(2πn)+12rsin(3πn)⇒sin(2πn)sin(3πn)=sin(3πn)sin(πn)+sin(2πn)sin(πn)⇒sin(2πn)[sin(3πn)−sin(πn)]=sin(3πn)sin(πn)Using transformation angle formula, we get⇒sin(2πn).2cos(2πn)sin(πn)=sin(3πn)sin(πn)⇒2sin(2πn)cos(2πn)=sin(3πn)Using multiple angle formula, 2sinAcosA=sin2A we getsin(4πn)=sin(3πn)∴4πn=r+(−1)r3n for r=1,n=7

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