Area of Triangle in Determinant Form with Vertex at Origin/Proof 1

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 $\AA$ of $T$ is:

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

Proof

\(\ds \map \Area T\)

\(=\)

\(\ds \dfrac 1 2 \size {\paren {\begin {vmatrix} 0 & 0 & 1 \\ b & a & 1 \\ x & y & 1 \end {vmatrix} } }\)

Area of Triangle in Determinant Form

\(\ds \)

\(=\)

\(\ds \dfrac 1 2 \size {b y - a x}\)

Determinant of Order 3

$\blacksquare$