# For a regular polygon, let $\dpi{100} r\; and\; R$ be the radii of the inscribed and the circumscribed circles. A false statement among the following is Option 1) there is a regular polygon with  $r/R=1/2\;$ Option 2) there is a regular polygon with  $\frac{r}{R}=\frac{1}{\sqrt{2}}\;$ Option 3) there is a regular polygon with  $\frac{r}{R}=\frac{2}{3}\;$ Option 4) there is a regular polygon with $\; \frac{r}{R}=\frac{\sqrt{3}}{2}$

V Vakul

As we learnt in

Relation between Systems of Measurement of Angles -

$\frac{D}{90} = \frac{G}{100} = \frac{2C}{\pi }$

- wherein

$\rightarrow$ Angle in Degrees

$\rightarrow$ Angle in Grades

$\rightarrow$ Angle in Radians

Let O be centre of Polygon, AB be any edge. In radius r = OC

Then R = OA = OB,   $\angle AOB =\frac{2\pi}{n}, \angle COB = \frac{\pi}{n}$ .                       AB = a

Then In $\Delta OCB, tan \angle COB=\frac{CB}{OC}\Rightarrow OC=\frac{BC}{tan(\angle COB)}=\frac{a}{2 tan\frac{\pi}{n}}$

In radius  $r= \frac{\alpha }{2}\ cot\ \frac{\pi}{n}\ \, \, \, \, \, \, \, ....(i)$

Similarly, In $\Delta COB, sin (\angle COB)=\frac{\alpha }{2\ OB} \Rightarrow OB = \frac{a}{2sin\frac{\pi }{n}}$

$R = \frac{\alpha }{2 \sin \frac{\pi }{n} }\ .....................(2)$

$\frac{r}{R}=cos\frac{\pi}{n}$        {divides (1) by (2)}

$(1)\ \frac{r}{R} = \frac{1 }{2} \Rightarrow \cos \frac{\pi }{n}=\frac{1}{2}\ \: \: \: \: \: \: \: \: \: \: \, \, \, \, \, possible \ for \ n = 3$

$(2)\ \frac{r}{R} = \frac{1 }{\sqrt{2}} \Rightarrow \cos \frac{{\pi }}{n}=\frac{1}{\sqrt{2}}\ \, \, \, \, \, \, \, \, \, \, \, \, \, possible\ for \ n = 4$

$(3) \frac{r}{R} = \frac{2 }{3} \Rightarrow \cos \frac{{\pi }}{n}=\frac{2}{3}\ \, \, \, \, \, \, \, \, \, \, \, \, \, \: \: \: \: \: \: \: \: \: not\ possible \ for \ an \ integer \ n$

$(4) \frac{r}{R} = \frac{\sqrt{3}}{2} \Rightarrow \cos \frac{\pi}{n}=\frac{\sqrt{3}}{2}\ \: \: \: \: \: \: \, \, \, possible\ for\ n = 6$

Option 1)

there is a regular polygon with  $r/R=1/2\;$

Incorrect

Option 2)

there is a regular polygon with  $\frac{r}{R}=\frac{1}{\sqrt{2}}\;$

Incorrect

Option 3)

there is a regular polygon with  $\frac{r}{R}=\frac{2}{3}\;$

Correct

Option 4)

there is a regular polygon with $\; \frac{r}{R}=\frac{\sqrt{3}}{2}$

Incorrect

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