Cho các điểm A(1; -2), B(-2; 3), C(0; 4). Tính diện tích tam giác ABC

Ta có: A(1; –2), B(–2; 3), C(0; 4)

\( \Rightarrow \left\{ {\begin{array}{*{20}{c}}{\overrightarrow {AB} = \left( { - 3;5} \right)}\\{\overrightarrow {BC} = \left( {2;1} \right)}\\{\overrightarrow {CA} = \left( {1; - 6} \right)}\end{array}} \right. \Rightarrow \left\{ {\begin{array}{*{20}{c}}{AB = \sqrt {{{\left( { - 3} \right)}^2} + {5^2}} = \sqrt {34} }\\{BC = \sqrt {{2^2} + {1^2}} = \sqrt 5 }\\{CA = \sqrt {{1^2} + {{\left( { - 6} \right)}^2}} = \sqrt {37} }\end{array}} \right.\)

p = \(\frac{{AB + BC + CA}}{2} = \frac{{\sqrt {34} + \sqrt 5 + \sqrt {37} }}{2}\)

\( \Rightarrow {S_{ABC}} = \sqrt {p\left( {p - AB} \right)\left( {p - BC} \right)\left( {p - CA} \right)} = \frac{{13}}{2}\).