a) In the accompanying figure, the area of the triangle ABC can be expressed as
area ABC = area ADEC + area CEFB - area ADFB
Use this and the fact the area of a trapezoid equals 1/2 the altitude times the sum of the parallel sides to show that:
(see attached filed)
Note: In the derivation of this formula, the vertices are labeled such that the triangle is traced coutnerclockwise proceeding from (x1, y1) to (x2, y2) to (x3, y3). For a clockwise orientation, the determinant above yields the negative of the area.
b) Use the result in (a) to find the triangle with vertices (3, 3), (4, 0), (-2, -1).
Area ABC = Area ADEC + Area CEFB - Area ADFB
Area ADEC = 1/2 (AD + EC)*DE = 1/2 (y1+y3)(x3 - x1)
Area CEFB = 1/2 (CE + FB)*EF = 1/2 (y3+y2)(x2 - x3)
Area ABDF = 1/2 (AD + BF)*DF = 1/2 (y1+y2)(x2 - ...
This shows how to use a matrix to find the area of a triangle.