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Question-186883




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=((sin 135 a)/(sin θ))  AC=((sin 75 (√2)a)/(sin (75+θ)))  ((sin (75+θ))/( (√2) sin 75))=((sin θ)/(sin 135))  (2−(√3))sin θ+cos θ=2 sin θ  tan θ=(1/( (√3))) ⇒θ=30°  ∠ACB=180−30−75=75=∠ABC  ⇒AC=AB
$${AC}=\frac{\mathrm{sin}\:\mathrm{135}\:{a}}{\mathrm{sin}\:\theta} \\ $$$${AC}=\frac{\mathrm{sin}\:\mathrm{75}\:\sqrt{\mathrm{2}}{a}}{\mathrm{sin}\:\left(\mathrm{75}+\theta\right)} \\ $$$$\frac{\mathrm{sin}\:\left(\mathrm{75}+\theta\right)}{\:\sqrt{\mathrm{2}}\:\mathrm{sin}\:\mathrm{75}}=\frac{\mathrm{sin}\:\theta}{\mathrm{sin}\:\mathrm{135}} \\ $$$$\left(\mathrm{2}−\sqrt{\mathrm{3}}\right)\mathrm{sin}\:\theta+\mathrm{cos}\:\theta=\mathrm{2}\:\mathrm{sin}\:\theta \\ $$$$\mathrm{tan}\:\theta=\frac{\mathrm{1}}{\:\sqrt{\mathrm{3}}}\:\Rightarrow\theta=\mathrm{30}° \\ $$$$\angle{ACB}=\mathrm{180}−\mathrm{30}−\mathrm{75}=\mathrm{75}=\angle{ABC} \\ $$$$\Rightarrow{AC}={AB} \\ $$
Commented by Mingma last updated on 12/Feb/23
Nice work!

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