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Question Number 191278 by Mingma last updated on 22/Apr/23

Answered by a.lgnaoui last updated on 23/Apr/23

chaque heptagone a 7 triangles isoceles de somet O  d angle 𝛉=((2Ο€)/7)  de cote x  avec    xsin (𝛉/2)=(1/2)  β‡’xsin (Ο€/7)=(1/2)    x=(1/(2sin (Ο€/7)))   du rayon du cercle circonscrit r=x  surface s=y.x   ;        [ y^2 +x^2 =((1/2))^2 ]  d apres graphe   y=(√((1/4)βˆ’(1/4)sin^2 ((𝛑/7))))                y=(1/2)cos (Ο€/7)  β‡’s=  (1/(2sin (Ο€/7)))Γ—(1/2)cos (Ο€/7)  =(1/(4tan (Ο€/7)))     totale  pour circonference:  =(7/(4tan (Ο€/7)))  totale  pour  les 14  circonferencds=                    Aire =((49)/(2tan(Ο€/7)))

$$\mathrm{chaque}\:\mathrm{heptagone}\:\mathrm{a}\:\mathrm{7}\:\mathrm{triangles}\:\mathrm{isoceles}\:\mathrm{de}\:\mathrm{somet}\:\mathrm{O} \\ $$$$\mathrm{d}\:\mathrm{angle}\:\boldsymbol{\theta}=\frac{\mathrm{2}\pi}{\mathrm{7}}\:\:\mathrm{de}\:\mathrm{cote}\:\boldsymbol{\mathrm{x}}\:\:\mathrm{avec}\:\:\:\:\boldsymbol{\mathrm{x}}\mathrm{sin}\:\frac{\boldsymbol{\theta}}{\mathrm{2}}=\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\Rightarrow\mathrm{xsin}\:\frac{\pi}{\mathrm{7}}=\frac{\mathrm{1}}{\mathrm{2}}\:\:\:\:\mathrm{x}=\frac{\mathrm{1}}{\mathrm{2sin}\:\frac{\pi}{\mathrm{7}}} \\ $$$$\:\mathrm{du}\:\mathrm{rayon}\:\mathrm{du}\:\mathrm{cercle}\:\mathrm{circonscrit}\:\boldsymbol{\mathrm{r}}=\boldsymbol{\mathrm{x}} \\ $$$$\boldsymbol{\mathrm{surface}}\:\boldsymbol{\mathrm{s}}=\boldsymbol{\mathrm{y}}.\boldsymbol{\mathrm{x}}\:\:\:;\:\:\:\:\:\:\:\:\left[\:\boldsymbol{\mathrm{y}}^{\mathrm{2}} +\boldsymbol{\mathrm{x}}^{\mathrm{2}} =\left(\frac{\mathrm{1}}{\mathrm{2}}\right)^{\mathrm{2}} \right] \\ $$$$\mathrm{d}\:\mathrm{apres}\:\mathrm{graphe}\:\:\:\boldsymbol{\mathrm{y}}=\sqrt{\frac{\mathrm{1}}{\mathrm{4}}βˆ’\frac{\mathrm{1}}{\mathrm{4}}\mathrm{sin}\:^{\mathrm{2}} \left(\frac{\boldsymbol{\pi}}{\mathrm{7}}\right)} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{y}}=\frac{\mathrm{1}}{\mathrm{2}}\mathrm{cos}\:\frac{\pi}{\mathrm{7}} \\ $$$$\Rightarrow\mathrm{s}=\:\:\frac{\mathrm{1}}{\mathrm{2sin}\:\frac{\pi}{\mathrm{7}}}Γ—\frac{\mathrm{1}}{\mathrm{2}}\mathrm{cos}\:\frac{\pi}{\mathrm{7}}\:\:=\frac{\mathrm{1}}{\mathrm{4tan}\:\frac{\pi}{\mathrm{7}}}\:\:\: \\ $$$$\mathrm{totale}\:\:\mathrm{pour}\:\mathrm{circonference}:\:\:=\frac{\mathrm{7}}{\mathrm{4tan}\:\frac{\pi}{\mathrm{7}}} \\ $$$$\mathrm{totale}\:\:\mathrm{pour}\:\:\mathrm{les}\:\mathrm{14}\:\:\mathrm{circonferencds}= \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{Aire}}\:=\frac{\mathrm{49}}{\mathrm{2}\boldsymbol{\mathrm{tan}}\frac{\pi}{\mathrm{7}}} \\ $$$$ \\ $$

Commented by a.lgnaoui last updated on 23/Apr/23

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