Properties Of Triangles Ques 49
$I _n$ is the area of $n$ sided regular polygon inscribed in a circle of unit radius and $O _n$ be the area of the polygon circumscribing the given circle, prove that
$ I _n=\frac{O _n}{2}(1+\sqrt{1-(\frac{2 I_n}{n}})^2) $
$(2003,5 M)$
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Solution:
- We know that, $I _n=\frac{n}{2} r^{2} \sin \frac{2 \pi}{n}$
[since, $I _n$ is area of regular polygon]
$\Rightarrow \frac{2 I _n}{n}=\sin \frac{2 \pi}{n} $ $\quad$ …….(i)
$\text { and } O _n=n r^{2} \tan \frac{\pi}{2}$
[since, $O _n$ is area of circumscribing polygon]
$ \frac{O _n}{n}=\tan \frac{\pi}{n} $ $\quad$ …….(ii)
On dividing Eq. (i) by Eq. (ii), we get
$ \frac{2 I _n}{O _n} =\frac{\sin \frac{2 \pi}{n}}{\tan \frac{\pi}{n}} $
$\Rightarrow \quad \frac{I _n}{O _n} =\cos ^{2} \frac{\pi}{n}=\frac{1+\cos \frac{2 \pi}{n}}{2} $
$\therefore \quad \frac{I _n}{O _n} =\frac{1+\sqrt{1-\left(2 I _n / n\right)^{2}}}{2} \quad \text { [from Eq. (i)] } $
$\Rightarrow \quad I _n =\frac{O _n}{2}\left(1+\sqrt{\left(1-\left(2 I _n / n\right)^{2}\right)}\right.$
BETA
