Question Number 203401 by cherokeesay last updated on 18/Jan/24
Answered by MM42 last updated on 18/Jan/24
$$<{B}=\mathrm{30}^{{o}} \\ $$$${AB}=\mathrm{2}\sqrt{\mathrm{3}}×{cos}\mathrm{30}=\mathrm{3} \\ $$$${R}=\sqrt{\mathrm{21}} \\ $$$${EC}=\mathrm{2}\sqrt{\mathrm{3}}×{cos}\mathrm{60}=\sqrt{\mathrm{3}} \\ $$$${r}=\sqrt{\mathrm{3}}×{tan}\mathrm{30}\Rightarrow{r}=\mathrm{1} \\ $$$$\Rightarrow\frac{{r}}{{R}}=\frac{\sqrt{\mathrm{21}}}{\mathrm{21}}\:\:\checkmark \\ $$$$ \\ $$
Commented by MM42 last updated on 18/Jan/24
Commented by cherokeesay last updated on 18/Jan/24
$${Nice}\:! \\ $$$${thank}\:{you} \\ $$
Commented by mr W last updated on 19/Jan/24
$${O}\:{doesn}'{t}\:{lie}\:{on}\:{CD}. \\ $$
Answered by mr W last updated on 18/Jan/24
Commented by mr W last updated on 18/Jan/24
$${R}^{\mathrm{2}} =\mathrm{3}^{\mathrm{2}} +\left(\mathrm{2}\sqrt{\mathrm{3}}\right)^{\mathrm{2}} =\mathrm{21} \\ $$$$\Rightarrow{R}=\sqrt{\mathrm{21}} \\ $$$$\left({R}−{r}\right)^{\mathrm{2}} ={r}^{\mathrm{2}} +\left(\sqrt{\mathrm{3}}+\sqrt{\mathrm{3}}{r}\right)^{\mathrm{2}} \\ $$$$\mathrm{3}{r}^{\mathrm{2}} +\mathrm{2}\left(\mathrm{3}+\sqrt{\mathrm{21}}\right){r}−\mathrm{18}=\mathrm{0} \\ $$$$\Rightarrow{r}=\frac{\sqrt{\mathrm{6}\left(\mathrm{14}+\sqrt{\mathrm{21}}\right)}−\mathrm{3}−\sqrt{\mathrm{21}}}{\mathrm{3}}\approx\mathrm{0}.\mathrm{9922} \\ $$$$\Rightarrow\frac{{r}}{{R}}=\frac{\sqrt{\mathrm{6}\left(\mathrm{14}+\sqrt{\mathrm{21}}\right)}−\mathrm{3}−\sqrt{\mathrm{21}}}{\mathrm{3}\sqrt{\mathrm{21}}}\approx\mathrm{0}.\mathrm{2165} \\ $$