Area of Triangle in Determinant Form with Vertex at Origin

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Example of Area of Triangle in Determinant Form

Let $A = \tuple {0, 0}, B = \tuple {b, a}, C = \tuple {x, y}$ be points in the Cartesian plane.

Let $T$ the triangle whose vertices are at $A$, $B$ and $C$.

Then the area $\mathcal A$ of $T$ is:

$\map \Area T = \dfrac {\size {b y - a x} } 2$


Proof

\(\displaystyle \map \Area T\) \(=\) \(\displaystyle \dfrac 1 2 \size {\paren {\begin{vmatrix} 0 & 0 & 1 \\ b & a & 1 \\ x & y & 1 \end{vmatrix} } }\) Area of Triangle in Determinant Form
\(\displaystyle \) \(=\) \(\displaystyle \dfrac 1 2 \size {b y - a x}\) Definition of Determinant of Order 3

$\blacksquare$


Example

Let $A = \tuple {0, 0}, B = \tuple {b, a}, C = \tuple {x_0, y_0}$ be points in the Cartesian plane.

Let $\tuple {x_0, y_0}$ be a solution to the linear diophantine equation:

$a x - b y = 1$

Let $T$ the triangle whose vertices are at $A$, $B$ and $C$.

Then the area $\mathcal A$ of $T$ is:

$\map \Area T = \dfrac 1 2$


Sources