Area of Triangle in Determinant Form with Vertex at Origin/Example

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Examples of Area of Triangle in Determinant Form with Vertex at Origin

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:

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


Proof

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

From Area of Triangle in Determinant Form with Vertex at Origin:

$\map \Area T = \dfrac 1 2 \size {b y - a x}$

for the triangle $T$ whose vertices are at $\tuple {0, 0}, \tuple {b, a}, \tuple {x, y}$.

We have that: $\size {b y_0 - a x_0} = 1$

Hence the result.

$\blacksquare$


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