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

  • 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}

 

Answers (1)

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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