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 $\AA$ 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.
$\blacksquare$
Sources
- 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {2-3}$ The Linear Diophantine Equation: Exercise $5$