Question

Let $G$ be a circle of radius $R>0$. Let $G_1,G_2,\dots ,G_n$ be $n$ circles of equal radius $r>0$. Suppose each of the $n$ circles $G_1,G_2,\dots ,G_n$ touches the circle $G$ externally. Also, for $i=1,2,\dots ,n-1$, the circle $G_i$ touches $G_{i+1}$ externally, and $G_n$ touches $G_1$ externally. Then, which of the following statements is/are TRUE ?

• A. If $n=4$, then $(\sqrt{2}-1)r<R$
• B. If $n=5$, then $r<R$
• C. If $n=8$, then $(\sqrt{2}-1)r<R$
• D. If $n=12$, then $\sqrt{2}(\sqrt{3}+1)r>R$

Answer 1

Answer: (C), (D)

$2(R+r)\sin \frac{\pi }{n}=2r$
$\frac{R+r}{r}=\mathrm{cosec}\frac{\pi }{n}$

(A) $n=4,R+r=\sqrt{2}r$
(B) $n=5,\frac{R+r}{r}=\mathrm{cosec}\frac{\pi }{5}<\mathrm{cosec}\frac{\pi }{6}$
$R+r<2r\Rightarrow r>R$
(C) $n=8,\frac{R+r}{r}=\mathrm{cosec}\frac{\pi }{8}>\mathrm{cosec}\frac{\pi }{4}$
$R+r>\sqrt{2}r$
(D) $n=12,\frac{R+r}{r}=\mathrm{cosec}\frac{\pi }{12}=\sqrt{2}(\sqrt{3}+1)$
$R+r=\sqrt{2}(\sqrt{3}+1)r$
$\sqrt{2}(\sqrt{3}+1)r>R$