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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

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

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

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