Question-191302

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

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

S_1 =S(OAB)=(1/2)AB×h_1 =((CD×h_1 )/4) (h_1 =OH_1 ) ((CD×h_1 )/4)=8 ⇒CD.h_1 =32 (1) S_2 =S(OCD)=(1/2)CD×h_2 =40 (h_2 =OH_2 ) ⇒ h_2 =((80)/(CD)) (2) S(Totale)=(((AB+CD))/2)×(h_1 +h_2 ) (3) (((1))/((2)))⇒ (h_1 /h_2 )=(2/5) ⇒h= h_1 +h_2 =(7/5)h_2 = h=((7×80)/(5×CD))=((112)/(CD)) (3)⇒ S=((3CD)/4)×((112)/(CD)) S(ABCD)= 84

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

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