Decomposing Bases for Area: Right Triangles via Rotation Setups

This example demonstrates how to decompose the area of a region using different “bases” (triangular arrangements) while preserving the same total area, highlighting the underlying invariance when the configuration changes. It centers on right-triangle area relationships, rigid transformations such as rotation (and the fact that area is preserved), and a distributive-style structure in area computation that can connect to quadratic-function reasoning. The construction is organized around a single main GeoGebra object (pic1), using geometric figures composed of triangles and their rotated counterparts to build multiple decompositions for comparison. In GeoGebra, the key steps are to create the relevant right-triangle pieces, generate matching configurations through rotation/transform tools, and use area measures/labels to verify that different base decompositions lead to the same area total. Suitable for middle school through high school geometry lessons, area proof practice, and integrating geometric reasoning with algebraic (quadratic) patterns.