Question Number 186883 by Mingma last updated on 11/Feb/23
Answered by HeferH last updated on 12/Feb/23
Commented by HeferH last updated on 12/Feb/23
Commented by Mingma last updated on 12/Feb/23
Great
Answered by mr W last updated on 12/Feb/23
Commented by mr W last updated on 12/Feb/23
$${AC}=\frac{\mathrm{sin}\:\mathrm{135}\:{a}}{\mathrm{sin}\:\theta} \\ $$$${AC}=\frac{\mathrm{sin}\:\mathrm{75}\:\sqrt{\mathrm{2}}{a}}{\mathrm{sin}\:\left(\mathrm{75}+\theta\right)} \\ $$$$\frac{\mathrm{sin}\:\left(\mathrm{75}+\theta\right)}{\:\sqrt{\mathrm{2}}\:\mathrm{sin}\:\mathrm{75}}=\frac{\mathrm{sin}\:\theta}{\mathrm{sin}\:\mathrm{135}} \\ $$$$\left(\mathrm{2}−\sqrt{\mathrm{3}}\right)\mathrm{sin}\:\theta+\mathrm{cos}\:\theta=\mathrm{2}\:\mathrm{sin}\:\theta \\ $$$$\mathrm{tan}\:\theta=\frac{\mathrm{1}}{\:\sqrt{\mathrm{3}}}\:\Rightarrow\theta=\mathrm{30}° \\ $$$$\angle{ACB}=\mathrm{180}−\mathrm{30}−\mathrm{75}=\mathrm{75}=\angle{ABC} \\ $$$$\Rightarrow{AC}={AB} \\ $$
Commented by Mingma last updated on 12/Feb/23
Nice work!