Coordinate Geometry: Area of Triangle with Vertex at Origin

Area of blue triangle OPQ = area of red rectangle minus the area of the outer three triangles.

= x₂y₁ – ½ [ (x₁ y₁ ) + (y₂ -y₁ )(x₁ -x₂ ) + (x₂ y₂ )]

= x₂y₁ – ½ [ x₁ y₁ + x₁ y₂ – x₂ y₂ – x₁y₁+ x₂ y₁ + x₂y₂]

= x₂y₁ – ½ [ x₁ y₂ + x₂ y₁ ]

= ½ [2 x₂y₁ – x₁ y₂ – x₂ y₁ ]

= ½ [x₂y₁ – x₁ y₂ ]

But since either P or Q could be designate (x₁y₁) or (x₂y₂) and the triangle can only have a positive area we use the absolute value of what is inside the square brackets to give the area of the blue triangle  = ½ | x₂y₁ – x₁ y₂ |.