Area of Triangle in Determinant Form/Examples/Vertices at (-4-i), (1+2i), (4-3i)

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Example of Area of Triangle in Determinant Form

Let $T$ be a triangle embedded in the complex plane with vertices at $\paren {-4 - i}, \paren {1 + 2 i}, \paren {4 - 3 i}$.

The area of $T$ is given by:

$\map \Area T = 17$


Proof

From Area of Triangle in Determinant Form:

$\map \Area T = \dfrac 1 2 \size {\paren {\begin{vmatrix} -4 & -1 & 1 \\ 1 & 2 & 1 \\ 4 & -3 & 1 \\ \end{vmatrix} } }$


\(\ds \map \Area T\) \(=\) \(\ds \dfrac 1 2 \size {\paren {\begin{vmatrix} -4 & -1 & 1 \\ 1 & 2 & 1 \\ 4 & -3 & 1 \\ \end{vmatrix} } }\)
\(\ds \) \(=\) \(\ds \dfrac 1 2 \size {\paren {\paren {-4} \times 2 - \paren {-1} \times 1} - \paren {\paren {-4} \times \paren {-3} - 4 \times \paren {-1} } + \paren {1 \times \paren {-3} - 2 \times 4} }\) Definition of Determinant
\(\ds \) \(=\) \(\ds \dfrac 1 2 \size {\paren {-8 - \paren {-1} } - \paren {12 - \paren {-4} } + \paren {-3 - 8} }\)
\(\ds \) \(=\) \(\ds \dfrac 1 2 \size {\paren {-7} - \paren {16} + \paren {-11} }\)
\(\ds \) \(=\) \(\ds \dfrac 1 2 \size {-34}\)
\(\ds \) \(=\) \(\ds 17\)

$\blacksquare$


Sources