\(\begin{array}{l}
{d_1}:\,\,\,y = 2x;\,\,{d_2}:\,\,\,y = \frac{1}{2}x;\,\,\,{d_3}:\,\,\,y = - x + 3\\
{d_3} \cap {d_1} = \left\{ A \right\};\,\,\,{d_3} \cap {d_2} = \left\{ B \right\}\\
a)\,\,\,Toa\,\,\,do\,\,\,diem\,\,\,A\,\,\,la\,\,\,nghiem\,\,\,cua\,\,he\,\,\,pt:\\
\left\{ \begin{array}{l}
y = - x + 3\\
y = 2x
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
x = - 3\\
y = - 6
\end{array} \right. \Rightarrow A\left( {1;\,\,2} \right)\\
Toa\,\,\,do\,\,\,diem\,\,\,B\,\,\,la\,\,\,nghiem\,\,\,cua\,\,he\,\,\,pt:\\
\left\{ \begin{array}{l}
y = - x + 3\\
y = \frac{1}{2}x
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
x = 2\\
y = 1
\end{array} \right. \Rightarrow B\left( {2;\,\,1} \right).\\
b)\,\,\,Ta\,\,co:\,\\
OA = \sqrt {{{\left( {{x_A} - {x_O}} \right)}^2} + {{\left( {{y_A} - {y_O}} \right)}^2}} = \sqrt {{1^2} + {2^2}} = \sqrt 5 .\\
OB = \sqrt {{{\left( {{x_B} - {x_O}} \right)}^2} + {{\left( {{y_B} - {y_O}} \right)}^2}} = \sqrt {{2^2} + {1^2}} = \sqrt 5 .\\
\Rightarrow OA = OB \Rightarrow \Delta AOB\,\,\,can\,\,tai\,\,\,O.\\
c)\,\,\,Ta\,\,\,co:\\
AB = \sqrt {{{\left( {{x_B} - {x_A}} \right)}^2} + {{\left( {{y_B} - {y_A}} \right)}^2}} = \sqrt {{{\left( {2 - 1} \right)}^2} + {{\left( {1 - 2} \right)}^2}} = \sqrt 2 \\
\Rightarrow Chu\,\,vi\,\,tam\,\,giac\,\,AOB\,\,la:\,\,\,\\
OA + OB + AB = 2\sqrt 5 + \sqrt 2 \,\,\,\,\,\left( {dvdd} \right).\\
Ke\,\,\,AH \bot Oy,\,\,\,BK\, \bot Ox\,\,\,nhu\,\,hinh\,\,ve.\\
Khi\,\,\,do\,\,ta\,\,co:\,\,\,\\
{S_{AOB}} = {S_{OMN}} - {S_{AOM}} - {S_{BON}}\\
= \frac{1}{2}OM.ON - \frac{1}{2}.AH.OM - \frac{1}{2}BK.ON\\
= \frac{1}{2}.3.3 - \frac{1}{2}.1.3 - \frac{1}{2}.3.1 = \frac{3}{2}\,\,\left( {dvdt} \right).
\end{array}\)
