Properties of Triangles 3 Question 16

16. In is the area of n sided regular polygon inscribed in a circle of unit radius and On be the area of the polygon circumscribing the given circle, prove that

In=On21+12In2n

(2003,5M)

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Answer:

Correct Answer: 16. 3

Solution:

  1. We know that, In=n2r2sin2πn

[since, In is area of regular polygon]

2Inn=sin2πn and On=nr2tanπ2

[since, On is area of circumscribing polygon]

Onn=tanπn

On dividing Eq. (i) by Eq. (ii), we get

2InOn=sin2πntanπnInOn=cos2πn=1+cos2πn2InOn=1+1(2In/n)22 [from Eq. (i)] In=On2(1+(1(2In/n)2)



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