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Question-193205

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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
β–³ABC ∑ABC=𝛂 ∑ACB=𝛃 ∑BAD=∑CAE=π›Œ ∑DAE=𝛉 β‡’2π›Œ+𝛉+𝛂+𝛃=180 β€’β–³ABC ((sin 𝛂)/y)=((sin 𝛃)/x) sin 𝛂=(y/x)sin 𝛃 (1) β€’β–³ABD et β–³ACD ((sin 𝛂)/(AD))=((sin π›Œ)/2) ((sin 𝛃)/(AD))=((sin 𝛉(+π›Œ))/8) β‡’((sin 𝛂)/(sin π›Œ))=((4sin 𝛃)/(sin (𝛉+π›Œ))) ((ysin 𝛃)/(xsin π›Œ))=((4sin 𝛃)/(sin (𝛉+π›Œ)))⇔ (y/(xsin π›Œ))=(4/(sin (𝛉+π›Œ))) (y/x)=((4sin π›Œ)/(sin (π›Œ+𝛉))) (2) β€’β–³ABE et β–³ACE ((sin 𝛂)/(AE))=((sin (π›Œ+𝛉))/5)=((sin (𝛃+π›Œ))/x) ((sin 𝛃)/(AE))=((sin π›Œ)/5) β‡’((5sin 𝛂)/(sin (π›Œ+𝛉)))=((5sin 𝛃)/(sin π›Œ))=((xsin 𝛂)/(sin (𝛃+π›Œ))) 1: ((sin 𝛂)/(sin (π›Œ+𝛉)))=((sin 𝛃)/(sin π›Œ)) β‡’sin 𝛃=((sin π›Œsin 𝛂)/(sin (π›Œ+𝛉)))=(y/(4x))sin 𝛂 ((sin 𝛃)/(sin 𝛂))=(y/(4x))= d apres (1) ((sin 𝛂)/(sin 𝛃))=(y/x)β‡’((sin 𝛃)/(sin 𝛂))=(x/y) β‡’ (y/(4x))=(x/y) y^2 =4x^2 soit (x/y)=(1/2)
$$\bigtriangleup\boldsymbol{\mathrm{ABC}}\:\:\:\:\measuredangle\boldsymbol{\mathrm{ABC}}=\boldsymbol{\alpha}\:\:\:\measuredangle\boldsymbol{\mathrm{ACB}}=\boldsymbol{\beta} \\ $$$$\measuredangle\boldsymbol{\mathrm{BAD}}=\measuredangle\boldsymbol{\mathrm{CAE}}=\boldsymbol{\lambda}\:\:\:\measuredangle\boldsymbol{\mathrm{DAE}}=\boldsymbol{\theta} \\ $$$$\Rightarrow\mathrm{2}\boldsymbol{\lambda}+\boldsymbol{\theta}+\boldsymbol{\alpha}+\boldsymbol{\beta}=\mathrm{180} \\ $$$$\bullet\bigtriangleup\boldsymbol{\mathrm{ABC}}\:\:\:\:\frac{\mathrm{sin}\:\boldsymbol{\alpha}}{\boldsymbol{\mathrm{y}}}=\frac{\mathrm{sin}\:\boldsymbol{\beta}}{\boldsymbol{\mathrm{x}}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{sin}\:\boldsymbol{\alpha}=\frac{\boldsymbol{\mathrm{y}}}{\boldsymbol{\mathrm{x}}}\mathrm{sin}\:\boldsymbol{\beta}\:\:\:\:\:\:\left(\mathrm{1}\right) \\ $$$$\bullet\bigtriangleup\boldsymbol{\mathrm{ABD}}\:\:\boldsymbol{\mathrm{et}}\:\bigtriangleup\boldsymbol{\mathrm{ACD}} \\ $$$$\:\:\frac{\mathrm{sin}\:\boldsymbol{\alpha}}{\boldsymbol{\mathrm{AD}}}=\frac{\mathrm{sin}\:\boldsymbol{\lambda}}{\mathrm{2}}\:\:\:\:\:\:\:\frac{\mathrm{sin}\:\boldsymbol{\beta}}{\boldsymbol{\mathrm{AD}}}=\frac{\mathrm{sin}\:\boldsymbol{\theta}\left(+\boldsymbol{\lambda}\right)}{\mathrm{8}} \\ $$$$\:\:\Rightarrow\frac{\mathrm{sin}\:\boldsymbol{\alpha}}{\mathrm{sin}\:\boldsymbol{\lambda}}=\frac{\mathrm{4sin}\:\boldsymbol{\beta}}{\mathrm{sin}\:\left(\boldsymbol{\theta}+\boldsymbol{\lambda}\right)} \\ $$$$\:\:\:\:\frac{\boldsymbol{\mathrm{y}}\mathrm{sin}\:\boldsymbol{\beta}}{\boldsymbol{\mathrm{x}}\mathrm{sin}\:\boldsymbol{\lambda}}=\frac{\mathrm{4sin}\:\boldsymbol{\beta}}{\mathrm{sin}\:\left(\boldsymbol{\theta}+\boldsymbol{\lambda}\right)}\Leftrightarrow\:\:\frac{\boldsymbol{\mathrm{y}}}{\boldsymbol{\mathrm{x}}\mathrm{sin}\:\boldsymbol{\lambda}}=\frac{\mathrm{4}}{\mathrm{sin}\:\left(\boldsymbol{\theta}+\boldsymbol{\lambda}\right)} \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\boldsymbol{\mathrm{y}}}{\boldsymbol{\mathrm{x}}}=\frac{\mathrm{4sin}\:\boldsymbol{\lambda}}{\mathrm{sin}\:\left(\boldsymbol{\lambda}+\boldsymbol{\theta}\right)}\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{2}\right) \\ $$$$ \\ $$$$\bullet\bigtriangleup\boldsymbol{\mathrm{ABE}}\:\:\:\boldsymbol{\mathrm{et}}\:\:\bigtriangleup\boldsymbol{\mathrm{ACE}} \\ $$$$\frac{\mathrm{sin}\:\boldsymbol{\alpha}}{\boldsymbol{\mathrm{AE}}}=\frac{\mathrm{sin}\:\left(\boldsymbol{\lambda}+\boldsymbol{\theta}\right)}{\mathrm{5}}=\frac{\mathrm{sin}\:\left(\boldsymbol{\beta}+\boldsymbol{\lambda}\right)}{\boldsymbol{\mathrm{x}}} \\ $$$$\frac{\mathrm{sin}\:\boldsymbol{\beta}}{\boldsymbol{\mathrm{AE}}}=\frac{\mathrm{sin}\:\boldsymbol{\lambda}}{\mathrm{5}} \\ $$$$\Rightarrow\frac{\mathrm{5sin}\:\boldsymbol{\alpha}}{\mathrm{sin}\:\left(\boldsymbol{\lambda}+\boldsymbol{\theta}\right)}=\frac{\mathrm{5sin}\:\boldsymbol{\beta}}{\mathrm{sin}\:\boldsymbol{\lambda}}=\frac{\mathrm{xsin}\:\boldsymbol{\alpha}}{\mathrm{sin}\:\left(\boldsymbol{\beta}+\boldsymbol{\lambda}\right)} \\ $$$$\mathrm{1}:\:\:\frac{\mathrm{sin}\:\boldsymbol{\alpha}}{\mathrm{sin}\:\left(\boldsymbol{\lambda}+\boldsymbol{\theta}\right)}=\frac{\mathrm{sin}\:\boldsymbol{\beta}}{\mathrm{sin}\:\boldsymbol{\lambda}}\:\: \\ $$$$\:\:\:\:\Rightarrow\mathrm{sin}\:\boldsymbol{\beta}=\frac{\mathrm{sin}\:\boldsymbol{\lambda}\mathrm{sin}\:\boldsymbol{\alpha}}{\mathrm{sin}\:\left(\boldsymbol{\lambda}+\boldsymbol{\theta}\right)}=\frac{\boldsymbol{\mathrm{y}}}{\mathrm{4}\boldsymbol{\mathrm{x}}}\mathrm{sin}\:\boldsymbol{\alpha} \\ $$$$\:\:\:\:\:\:\frac{\mathrm{sin}\:\boldsymbol{\beta}}{\mathrm{sin}\:\boldsymbol{\alpha}}=\frac{\boldsymbol{\mathrm{y}}}{\mathrm{4}\boldsymbol{\mathrm{x}}}= \\ $$$$\boldsymbol{\mathrm{d}}\:\boldsymbol{\mathrm{apres}}\:\left(\mathrm{1}\right)\:\frac{\mathrm{sin}\:\boldsymbol{\alpha}}{\mathrm{sin}\:\boldsymbol{\beta}}=\frac{\boldsymbol{\mathrm{y}}}{\boldsymbol{\mathrm{x}}}\Rightarrow\frac{\mathrm{sin}\:\boldsymbol{\beta}}{\mathrm{sin}\:\boldsymbol{\alpha}}=\frac{\boldsymbol{\mathrm{x}}}{\boldsymbol{\mathrm{y}}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\Rightarrow\:\:\frac{\boldsymbol{\mathrm{y}}}{\mathrm{4}\boldsymbol{\mathrm{x}}}=\frac{\boldsymbol{\mathrm{x}}}{\boldsymbol{\mathrm{y}}}\:\:\:\boldsymbol{\mathrm{y}}^{\mathrm{2}} =\mathrm{4}\boldsymbol{\mathrm{x}}^{\mathrm{2}} \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{soit}\:\:\:\:\frac{\boldsymbol{\mathrm{x}}}{\boldsymbol{\mathrm{y}}}=\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\:\:\:\:\:\:\: \\ $$$$ \\ $$

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