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Question Number 141909 by mathdanisur last updated on 24/May/21

Commented by mr W last updated on 24/May/21

$$BC=\frac{5}{2}$$

Answered by mr W last updated on 24/May/21

$$AB=h$$$$BC=x$$$$\tan \alpha =\frac{x}{h}$$$$\tan 2\alpha =\frac{x+4}{h}=\frac{2\times \frac{x}{h}}{1-\frac{x^2}{h^2}}$$$$\Rightarrow h^2=\frac{x^2(x+4)}{4-x}$$$$\tan 3\alpha =\frac{x+20}{h}=\frac{\frac{x}{h}(3-\frac{x^2}{h^2})}{1-3\times \frac{x^2}{h^2}}$$$$\Rightarrow h^2=\frac{x^2(x+30)}{10-x}$$$$\Rightarrow \frac{x^2(x+30)}{10-x}=\frac{x^2(x+4)}{4-x}$$$$\Rightarrow \frac{x+30}{10-x}=\frac{x+4}{4-x}$$$$\Rightarrow x=\frac{5}{2}$$

Commented by mathdanisur last updated on 25/May/21

$$coolSirthanks$$

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