Area Derivation - Triangle, Three Vertices

Category: Analytic Geometry, Plane Geometry, Algebra

"Published in Newark, California, USA"

Given a triangle with vertices as shown below

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Prove that the area of a triangle is

Solution:

The first step is to draw vertical lines from the vertices to the x-axis as shown below

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Area of Triangle ABC = Area of Trapezoid ACDF - Area of Trapezoid ABEF - Area of Trapezoid BCDE

Label further the figure, we have

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Consider Trapezoid ACDF

Consider Trapezoid ABEF

Consider Trapezoid BCDE

Therefore

Using a distance of two points formula

The above equation becomes

You notice that the above equation looks like a matrix or determinant because of the sequence of x and y. The three positive terms are the principal diagonals (product from top left to bottom right) while the three negative terms are the secondary diagonals (product from bottom left to top right). The above equation can be written as

Therefore