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JEE Main 2010
Exam
JEE Main
Year
2010
Subject
Mathematics
Chapter
Properties of Triangles

JEE Main 2010 · Mathematics · Properties of Triangles

For a regular polygon, let r and R be the radii of the inscribed...

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

Options

  1. AThere is a regular polygon with rR=12\frac{r}{R}=\frac{1}{\sqrt{2}}
  2. BThere is a regular polygon with rR=23\frac{r}{R}=\frac{2}{3}
  3. CThere is a regular polygon with rR=32\frac{r}{R}=\frac{\sqrt{3}}{2}
  4. DThere is a regular polygon with rR=12\frac{r}{R}=\frac{1}{2}

Correct answer

B. There is a regular polygon with rR=23\frac{r}{R}=\frac{2}{3}

Step-by-step solution

\mathrm{r}=\frac{\mathrm{a}}{2} \cot \frac{\pi}{\mathrm{n}}′a′issideofpolygon.'a' is side of polygon.R=a2cosec⁡πnrR=cot⁡πncosec⁡πn=cos⁡πncos⁡πn≠23 for any n∈N\begin{aligned}& R=\frac{a}{2} \operatorname{cosec} \frac{\pi}{n}\\ & \frac{r}{R}=\frac{\cot \frac{\pi}{n}}{\operatorname{cosec} \frac{\pi}{n}}=\cos \frac{\pi}{n}\\ & \cos \frac{\pi}{n} \neq \frac{2}{3} \text{ for any }n \in N \end{aligned}$
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