Plane Geometry - Bruce E. Shapiro

188 SECTION 37. AREADefinition 37.5 A Triangulation of a polygonal region is a completedecomposition of the region into non-overlapping triangular regions suchthat every point in the the region is contained in at least one triangularregion.Axiom 37.6 (Neutral Area Postulate) Associate with each polygonalregion R there is a nonnegative number α(R) called the area of R such that1. (Congruence) If two triangles are congruent then their associated areasare equal.2. (Additivity) If R is the union of two non-overlapping polygonal regionsR 1 and R 2 then α(R) = α(R 1 ) + α(R 2 ).Definition 37.7 The Rectangular Region ABCD is the union of thetriangular regions of the four triangles formed by the intersecting diagonalsof the rectangle:ABCD =AED ∪ DEC∪ CEB ∪ BEAwhere E is the intersection of the diagonals.We also definelength(R) = ABwidth(R) = BCFigure 37.2: A rectangular region is the union of the four triangular regionsdefined by the edges two intersecting diagonals.Axiom 37.8 (Euclidean Area Postulate) Let R be a rectangle. Thenα(R) = length(R) × width(R)Theorem 37.9 Let T = ABC where △ABC is a right triangle with rightangle at C. Thenα(T ) = 1 AC × BC2« CC BY-NC-ND 3.0. Revised: 18 Nov 2012