Question Number 205588 by cortano12 last updated on 25/Mar/24
Answered by A5T last updated on 25/Mar/24
Commented by A5T last updated on 25/Mar/24
$${BC}={r}\sqrt{\mathrm{2}};{BD}^{\mathrm{2}} ={r}^{\mathrm{2}} +{r}^{\mathrm{2}} −\mathrm{2}{r}^{\mathrm{2}} {cos}\mathrm{60}\Rightarrow{BD}={r} \\ $$$$\mathrm{6}^{\mathrm{2}} +{AB}^{\mathrm{2}} −\mathrm{6}\sqrt{\mathrm{3}}{AB}={r}^{\mathrm{2}} …\left({i}\right)\Rightarrow\mathrm{2}{r}^{\mathrm{2}} =\mathrm{72}+\mathrm{2}{AB}^{\mathrm{2}} −\mathrm{12}\sqrt{\mathrm{3}}{AB} \\ $$$$\mathrm{2}{r}^{\mathrm{2}} ={BC}^{\mathrm{2}} =\mathrm{96}+{AB}^{\mathrm{2}} −\mathrm{8}\sqrt{\mathrm{3}}{AB}=\mathrm{72}+\mathrm{2}{AB}^{\mathrm{2}} −\mathrm{12}\sqrt{\mathrm{3}}{AB} \\ $$$$\Rightarrow{AB}^{\mathrm{2}} +\mathrm{4}\sqrt{\mathrm{3}}{AB}−\mathrm{24}=\mathrm{0}\Rightarrow{AB}=\mathrm{2}\left(\sqrt{\mathrm{3}}+\mathrm{3}\right)\approx\mathrm{9}.\mathrm{464} \\ $$
Answered by mr W last updated on 25/Mar/24
Commented by mr W last updated on 25/Mar/24
$${CD}^{\mathrm{2}} =\mathrm{6}^{\mathrm{2}} +\left(\mathrm{4}\sqrt{\mathrm{6}}\right)^{\mathrm{2}} −\mathrm{2}×\mathrm{6}×\mathrm{4}\sqrt{\mathrm{6}}×\mathrm{cos}\:\mathrm{75}° \\ $$$$\Rightarrow{CD}=\mathrm{2}\sqrt{\mathrm{15}+\mathrm{6}\sqrt{\mathrm{3}}} \\ $$$$\frac{\mathrm{4}\sqrt{\mathrm{6}}}{\mathrm{sin}\:\alpha}=\frac{\mathrm{2}\sqrt{\mathrm{15}+\mathrm{6}\sqrt{\mathrm{3}}}}{\mathrm{sin}\:\mathrm{75}°}=\mathrm{2}{R} \\ $$$$\mathrm{sin}\:\alpha=\frac{\mathrm{4}\sqrt{\mathrm{6}}×\mathrm{sin}\:\mathrm{75}°}{\mathrm{2}\sqrt{\mathrm{15}+\mathrm{6}\sqrt{\mathrm{3}}}}=\frac{\mathrm{1}+\sqrt{\mathrm{3}}}{\:\sqrt{\mathrm{5}+\mathrm{2}\sqrt{\mathrm{3}}}} \\ $$$$\Rightarrow\mathrm{cos}\:\alpha=\frac{\mathrm{1}}{\:\sqrt{\mathrm{5}+\mathrm{2}\sqrt{\mathrm{3}}}} \\ $$$$\frac{{AB}}{\mathrm{sin}\:\left(\alpha+\mathrm{45}°\right)}=\mathrm{2}{R} \\ $$$$\Rightarrow{AB}=\frac{\mathrm{2}\sqrt{\mathrm{15}+\mathrm{6}\sqrt{\mathrm{3}}}}{\mathrm{sin}\:\mathrm{75}°}×\frac{\mathrm{sin}\:\alpha+\mathrm{cos}\:\alpha}{\:\sqrt{\mathrm{2}}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{2}\left(\mathrm{3}+\sqrt{\mathrm{3}}\right)\:\checkmark \\ $$