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Question Number 82901 by Power last updated on 25/Feb/20

Answered by mr W last updated on 26/Feb/20

Commented by Power last updated on 26/Feb/20

$$\mathrm{sir}\mathrm{a}:\mathrm{b}:\mathrm{c}=?$$

Commented by mr W last updated on 26/Feb/20

$$letBC=1$$$$a=\sin 30=\frac{1}{2}$$$$\frac{CF}{\sin 60}=\frac{1}{\sin 45}$$$$\Rightarrow CF=\frac{\sin 60}{\sin 45}=\frac{\sqrt{6}}{2}$$$$AC=\frac{1}{2\cos 75}=\frac{2}{\sqrt{6}-\sqrt{2}}=\frac{\sqrt{6}+\sqrt{2}}{2}$$$$AF=\frac{\sqrt{6}+\sqrt{2}}{2}-\frac{\sqrt{6}}{2}=\frac{\sqrt{2}}{2}$$$$EF=\frac{\sqrt{2}}{2\sqrt{3}}=\frac{\sqrt{6}}{6}$$$$\tan \alpha =\frac{\sqrt{6}}{6\times \frac{\sqrt{6}}{2}}=\frac{1}{3}$$$$HF=b+c=CH=\cos 30=\frac{\sqrt{3}}{2}$$$$b=HC\times \tan \beta =\frac{\sqrt{3}}{2}\times \tan (45-\alpha )$$$$=\frac{\sqrt{3}}{2}\times \frac{1-\frac{1}{3}}{1+\frac{1}{3}}=\frac{\sqrt{3}}{4}$$$$c=\frac{\sqrt{3}}{2}-\frac{\sqrt{3}}{4}=\frac{\sqrt{3}}{4}$$$$\Rightarrow a=\frac{1}{2},b=c=\frac{\sqrt{3}}{4}$$$$\Rightarrow a:b:c=2:\sqrt{3}:\sqrt{3}$$

Commented by Power last updated on 26/Feb/20

$$\mathrm{thanks}$$

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