Bonsoir,
[tex]a)[/tex]
[tex]\textnormal{D'apr\`es le th\'eor\`eme de Pythagore : }[/tex]
[tex]EC^2 = EB^2 + BC^2 \iff EC^2 = (\frac{a}{2})^2 + a^2\iff EC^2= \frac{a^2}{4} + a^2[/tex]
[tex]EC^2= \frac{a^2}{4} + a^2 \iff EC^2= \frac{a^2+ 4a^2}{4} \iff EC^2= \frac{5a^2}{4}[/tex]
[tex]EC = \sqrt{\frac{5a^2}{4} } \iff \boxed{EC = \frac{\sqrt{5} a}{2} }[/tex]
[tex]b)[/tex]
[tex]EC = BF[/tex]
[tex]AF = \frac{a}{2} + \frac{\sqrt{5}a }{2} = \frac{a+\sqrt{5}a }{2}[/tex]
[tex]\frac{AF}{FG} = \frac{\frac{a+\sqrt{5}a }{2} }{a} \iff \frac{a+\sqrt{5}a}{2} \times \frac{1}{a} \iff \frac{a + \sqrt{5}a }{2a} \iff \frac{a(1+\sqrt{5}) }{2a} \iff \boxed{\frac{1+\sqrt{5} }{2} } = \phi[/tex]
