Đáp án:
$\begin{array}{l}
a)AC = \sqrt {{{\left( {3 - 1} \right)}^2} + {{\left( { - 3 - 5} \right)}^2}} = 2\sqrt {17} \\
OA = \sqrt {{3^2} + {{\left( { - 3} \right)}^2}} = 3\sqrt 2 \\
OC = \sqrt {{1^2} + {5^2}} = \sqrt {26} \\
\Rightarrow Chu\,vi:\Delta OAC = 2\sqrt {17} + 3\sqrt 2 + \sqrt {26} \\
b)\overrightarrow {AO} = \left( { - 3;3} \right);\overrightarrow {CO} = \left( { - 1; - 5} \right)\\
\Rightarrow \overrightarrow {AO} .\overrightarrow {CO} = - 3.\left( { - 1} \right) + 3.\left( { - 5} \right) = - 12\\
\Rightarrow cos\left( {\overrightarrow {AO} ;\overrightarrow {CO} } \right) = \dfrac{{\overrightarrow {AO} .\overrightarrow {CO} }}{{AO.CO}}\\
= \dfrac{{ - 12}}{{3\sqrt 2 .\sqrt {26} }} = \dfrac{{ - 2\sqrt {13} }}{{13}}\\
c)H\left( {x;y} \right)\\
H \in AC;{S_{OAH}} = 2.{S_{OCH}}\\
\Rightarrow AH = 2.CH\\
\Rightarrow \left[ \begin{array}{l}
\overrightarrow {AH} = 2.\overrightarrow {CH} \\
\overrightarrow {AH} = 2.\overrightarrow {HC}
\end{array} \right.\\
\Rightarrow \left[ \begin{array}{l}
\left( {x - 3;y + 3} \right) = 2.\left( {x - 1;y - 5} \right)\\
\left( {x - 3;y + 3} \right) = 2.\left( {1 - x;5 - y} \right)
\end{array} \right.\\
\Rightarrow \left[ \begin{array}{l}
\left\{ \begin{array}{l}
x - 3 = 2\left( {x - 1} \right)\\
y + 3 = 2.\left( {y - 5} \right)
\end{array} \right.\\
\left\{ \begin{array}{l}
x - 3 = 2\left( {1 - x} \right)\\
y + 3 = 2.\left( {5 - y} \right)
\end{array} \right.
\end{array} \right.\\
\Rightarrow \left[ \begin{array}{l}
x = - 1;y = 13\\
x = \dfrac{5}{3};y = \dfrac{7}{3}
\end{array} \right.\\
\Rightarrow \left[ \begin{array}{l}
H\left( { - 1;13} \right)\\
H\left( {\dfrac{5}{3};\dfrac{7}{3}} \right)
\end{array} \right.
\end{array}$
