Question

Dau coroniță!
Cât mai complex ​

Answer 1

$\displaystyle\bf\\\mathcal{P}_{ADC}=34cm \Leftrightarrow AD~+~DC~+~AC=34cm.\\\mathcal{P}_{AOB}=24cm \Leftrightarrow AO~+~OB~+~AB~=24cm\\si~cum~ABCD~este~patrat~\implies~AO=OC=OB=OD=x.\\analog \implies AB=BC=DC=AD=y.\\asadar,~acum~inlocuim~:\\ 2y+2x = 34,~(AC~este~AO~+~OC=2x)\\2x+y=24,~(AO~+~OB~=2x)\\rezolvand~sistemul~obtinem~ca~x=7,~y=10.\\deci : AO=OC=OB=OD=7,~si~AB=BC=DC=AD=10.\\\mathcal{A}_{ABCD}=10^2=100cm^2.\\cum~BO=DO=7 \implies BM=OM=ON=ND=\frac{7}{2}.$

$\displaystyle\bf\\\mathcal{A}_{AMCN}=2\mathcal{A}_{AON}+2\mathcal{A}_{CON}=4\mathcal{A}_{AON}.\\\mathcal{A}_{AON}=\frac{AO\cdot AN}{2} =\frac{7\cdot \frac{7}{2} }{2}=\frac{49}{4} cm^2.\\\mathcal{A}_{AMCN}=\frac{49}{4} \cdot 4= 49cm^2.\\pentru~ca~:\mathcal{A}_{AON}=\mathcal{A}_{AOM}=\mathcal{A}_{OMC} =\mathcal{A}_{ONC}.$