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Question Number 203401 by cherokeesay last updated on 18/Jan/24

Answered by MM42 last updated on 18/Jan/24

$$<B={30}^o$$$$AB=2\sqrt{3}\times cos30=3$$$$R=\sqrt{21}$$$$EC=2\sqrt{3}\times cos60=\sqrt{3}$$$$r=\sqrt{3}\times tan30\Rightarrow r=1$$$$\Rightarrow \frac{r}{R}=\frac{\sqrt{21}}{21}✓$$$$$$

Commented by MM42 last updated on 18/Jan/24

Commented by cherokeesay last updated on 18/Jan/24

$$Nice!$$$$thankyou$$

Commented by mr W last updated on 19/Jan/24

$$O{doesn}^{\prime }tlieonCD.$$

Answered by mr W last updated on 18/Jan/24

Commented by mr W last updated on 18/Jan/24

$$R^2=3^2+{\left(2\sqrt{3}\right)}^2=21$$$$\Rightarrow R=\sqrt{21}$$$${(R-r)}^2=r^2+{(\sqrt{3}+\sqrt{3}r)}^2$$$$3r^2+2(3+\sqrt{21})r-18=0$$$$\Rightarrow r=\frac{\sqrt{6(14+\sqrt{21})}-3-\sqrt{21}}{3}\approx 0.9922$$$$\Rightarrow \frac{r}{R}=\frac{\sqrt{6(14+\sqrt{21})}-3-\sqrt{21}}{3\sqrt{21}}\approx 0.2165$$

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