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Question

Let A1,A2,....An be the vertices of an nsided 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 7
Let 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(π3πn)
4πn=π3πn
7πn=π
Thus, n=7

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