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Question Number 191302 by Mingma last updated on 22/Apr/23

Answered by a.lgnaoui last updated on 22/Apr/23

$${\mathbf{S}}_1=\mathbf{S}\left(\mathbf{OAB}\right)=\frac{1}{2}\mathrm{AB}\times {\mathrm{h}}_1=\frac{\mathrm{CD}\times {\mathrm{h}}_1}{4}({\mathrm{h}}_1={\mathrm{OH}}_1)$$$$\frac{\mathrm{CD}\times {\mathrm{h}}_1}{4}=8\Rightarrow \mathbf{CD}.{\mathbf{h}}_1=32\left(1\right)$$$${\mathbf{S}}_2=\mathbf{S}\left(\mathbf{OCD}\right)=\frac{1}{2}\mathrm{CD}\times {\mathrm{h}}_2=40({\mathrm{h}}_2={\mathrm{OH}}_2)$$$$\Rightarrow {\mathbf{h}}_2=\frac{80}{\mathrm{CD}}\left(2\right)$$$$$$$$\mathbf{S}\left(\mathbf{Totale}\right)=\frac{(\mathrm{AB}+\mathrm{CD})}{2}\times ({\mathrm{h}}_1+{\mathrm{h}}_2)\left(3\right)$$$$\frac{\left(1\right)}{\left(2\right)}\Rightarrow \frac{{\mathbf{h}}_1}{{\mathbf{h}}_2}=\frac{2}{5}\Rightarrow \mathbf{h}={\mathbf{h}}_1+{\mathbf{h}}_2=\frac{7}{5}{\mathbf{h}}_2=$$$$\mathrm{h}=\frac{7\times 80}{5\times \mathrm{CD}}=\frac{112}{\mathbf{CD}}$$$$\left(3\right)\Rightarrow \mathbf{S}=\frac{3\mathrm{CD}}{4}\times \frac{112}{\mathrm{CD}}$$$$$$$$\mathbf{S}\left(\mathbf{ABCD}\right)=84$$$$$$

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