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# For a regular polygon, let r and R be the radii of the inscribed and the circumscribed circles. A false statement among the following is

$\begin {array} {1 1} (A)\;there \: is \: a \: regular\: polygon\: with\: \large\frac{r}{R}=\large\frac{1}{2} \\ (B)\;there\: is\: a \: regular\: polygon\: with\: \large\frac{ r}{R} =\large\frac{1}{\sqrt 2} \\ (C)\;there\: is\: a\: regular\: polygon \: with \: \large\frac{r}{R}=\large\frac{2}{3} \\ (D)\;there\: is\: a\: regular\: polygon \: with\: \large\frac{ r}{R}=\large\frac{ \sqrt3}{2} \end {array}$

For diagram 1 $A=\large\frac{2 \pi}{n}$
$\large\frac{a}{2R}=\large\frac{ \sin\: \pi}{n}$
For diagram 2
$A=\large\frac{\pi}{n}$
$\large\frac{a}{2r}=\large\frac{ \tan\: \pi}{n}$
So,$\large\frac{r}{R}= \large\frac{ \cos\: \pi}{n}$
$n=3\: \: \: gives\: \: \large\frac{r}{R}=\large\frac{1}{2}$
$n=4\: \: \: gives\: \: \large\frac{r}{R}=\large\frac{1}{\sqrt 2}$
$n=6\: \: \: gives\: \: \large\frac{r}{R}=\large\frac{\sqrt 3}{2}$
edited Mar 26, 2014