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Question Number 193205 by Rupesh123 last updated on 07/Jun/23

Commented by a.lgnaoui last updated on 08/Jun/23

Answered by a.lgnaoui last updated on 07/Jun/23

$$△\mathbf{ABC}\measuredangle \mathbf{ABC}=\mathit{\alpha }\measuredangle \mathbf{ACB}=\mathit{\beta }$$$$\measuredangle \mathbf{BAD}=\measuredangle \mathbf{CAE}=\mathit{\lambda }\measuredangle \mathbf{DAE}=\mathit{\theta }$$$$\Rightarrow 2\mathit{\lambda }+\mathit{\theta }+\mathit{\alpha }+\mathit{\beta }=180$$$$\bullet △\mathbf{ABC}\frac{\sin \mathit{\alpha }}{\mathbf{y}}=\frac{\sin \mathit{\beta }}{\mathbf{x}}$$$$\sin \mathit{\alpha }=\frac{\mathbf{y}}{\mathbf{x}}\sin \mathit{\beta }\left(1\right)$$$$\bullet △\mathbf{ABD}\mathbf{et}△\mathbf{ACD}$$$$\frac{\sin \mathit{\alpha }}{\mathbf{AD}}=\frac{\sin \mathit{\lambda }}{2}\frac{\sin \mathit{\beta }}{\mathbf{AD}}=\frac{\sin \mathit{\theta }(+\mathit{\lambda })}{8}$$$$\Rightarrow \frac{\sin \mathit{\alpha }}{\sin \mathit{\lambda }}=\frac{4\sin \mathit{\beta }}{\sin (\mathit{\theta }+\mathit{\lambda })}$$$$\frac{\mathbf{y}\sin \mathit{\beta }}{\mathbf{x}\sin \mathit{\lambda }}=\frac{4\sin \mathit{\beta }}{\sin (\mathit{\theta }+\mathit{\lambda })}\iff \frac{\mathbf{y}}{\mathbf{x}\sin \mathit{\lambda }}=\frac{4}{\sin (\mathit{\theta }+\mathit{\lambda })}$$$$$$$$\frac{\mathbf{y}}{\mathbf{x}}=\frac{4\sin \mathit{\lambda }}{\sin (\mathit{\lambda }+\mathit{\theta })}\left(2\right)$$$$$$$$\bullet △\mathbf{ABE}\mathbf{et}△\mathbf{ACE}$$$$\frac{\sin \mathit{\alpha }}{\mathbf{AE}}=\frac{\sin (\mathit{\lambda }+\mathit{\theta })}{5}=\frac{\sin (\mathit{\beta }+\mathit{\lambda })}{\mathbf{x}}$$$$\frac{\sin \mathit{\beta }}{\mathbf{AE}}=\frac{\sin \mathit{\lambda }}{5}$$$$\Rightarrow \frac{5\sin \mathit{\alpha }}{\sin (\mathit{\lambda }+\mathit{\theta })}=\frac{5\sin \mathit{\beta }}{\sin \mathit{\lambda }}=\frac{\mathrm{xsin}\mathit{\alpha }}{\sin (\mathit{\beta }+\mathit{\lambda })}$$$$1:\frac{\sin \mathit{\alpha }}{\sin (\mathit{\lambda }+\mathit{\theta })}=\frac{\sin \mathit{\beta }}{\sin \mathit{\lambda }}$$$$\Rightarrow \sin \mathit{\beta }=\frac{\sin \mathit{\lambda }\sin \mathit{\alpha }}{\sin (\mathit{\lambda }+\mathit{\theta })}=\frac{\mathbf{y}}{4\mathbf{x}}\sin \mathit{\alpha }$$$$\frac{\sin \mathit{\beta }}{\sin \mathit{\alpha }}=\frac{\mathbf{y}}{4\mathbf{x}}=$$$$\mathbf{d}\mathbf{apres}\left(1\right)\frac{\sin \mathit{\alpha }}{\sin \mathit{\beta }}=\frac{\mathbf{y}}{\mathbf{x}}\Rightarrow \frac{\sin \mathit{\beta }}{\sin \mathit{\alpha }}=\frac{\mathbf{x}}{\mathbf{y}}$$$$\Rightarrow \frac{\mathbf{y}}{4\mathbf{x}}=\frac{\mathbf{x}}{\mathbf{y}}{\mathbf{y}}^2=4{\mathbf{x}}^2$$$$$$$$\mathrm{soit}\frac{\mathbf{x}}{\mathbf{y}}=\frac{1}{2}$$$$$$$$$$

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