Properties of Triangles 1 Question 19

19. Let A1,A2,,An be the vertices of an n-sided regular polygon such that 1A1A2=1A1A3+1A1A4. Find the value of n.

(1994,4 M)

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Answer:

Correct Answer: 19. 2

A=75

Solution:

  1. Let O be the centre and r be the radius of the circle passing through the vertices A1,A2,,An.

Then,

A1OA2=2πnOA1=OA2=r

also

Again, by cos formula, we know that,

cos2πn=OA12+OA22A1A222(OA1)(OA2)

cos2πn=r2+r2A1A222(r)(r)2r2cos2πn=2r2A1A22A1A22=2r22r2cos2πnA1A22=2r21cos2πnA1A22=2r22sin2πnA1A22=4r2sin2πnA1A2=2rsinπn Similarly, A1A3=2rsin2πn and A1A4=2rsin3πn Since, 1A1A2=1A1A3+1A1A4 [given] 12rsin(π/n)=12rsin(2π/n)+12rsin(3π/n)1sin(π/n)=1sin(2π/n)+1sin(3π/n)1sin(π/n)=sin3πn+sin2πnsin(2π/n)sin(3π/n)sin2πnsin3πn=sinπnsin3πn+sinπnsin2πnsin2πnsin3πnsinπn=sinπnsin3πnsin2πn2cos3π+π2nsin3ππ2n=sinπnsin3πn2sin2πncos2πnsinπn=sinπnsin3πn2sin2πncos2πn=sin3πnsin4πn=sin3πn4πn=π3πn7πn=πn=7



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