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Question Number 205681 by naka3546 last updated on 27/Mar/24

Answered by mr W last updated on 27/Mar/24

$$a=1cm$$$$R=\frac{1}{3}\times \frac{\sqrt{3}a}{2}=\frac{\sqrt{3}a}{6}$$$$r=\frac{\sqrt{3}b}{6}$$$$\frac{b}{a}=\frac{\frac{\sqrt{3}a}{2}-2R}{\frac{\sqrt{3}a}{2}}=\frac{1}{3}$$$$\Rightarrow r=\frac{\sqrt{3}}{6}\times \frac{a}{3}=\frac{\sqrt{3}a}{18}=\frac{\sqrt{3}}{18}$$

Commented by naka3546 last updated on 27/Mar/24

$$\mathrm{Thank}\mathrm{you},\mathrm{sir}.$$

Answered by A5T last updated on 27/Mar/24

Commented by A5T last updated on 27/Mar/24

$$a+b=a+a\Rightarrow a=b\Rightarrow 3a=\frac{1}{2}\Rightarrow a=\frac{1}{6}$$$$\Rightarrow \frac{r}{R}=\frac{a+b=2a}{1}=\frac{1}{3}\Rightarrow r=\frac{R}{3}$$$$R\left(\frac{3}{2}\right)=\frac{\sqrt{3}}{4}\Rightarrow R=\frac{\sqrt{3}}{6}\Rightarrow r=\frac{\sqrt{3}}{18}$$

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