Exploring Rectangles: Dynamic Geometry and Area Integration

This GeoGebra example dynamically explores the properties of rectangles, including side lengths, angles, diagonals, and the relationship between area and definite integrals. The core mathematical concepts involve the definition of a rectangle (four right angles, opposite sides parallel and equal), translation transformations, definite integrals for area calculation, and the auxiliary use of scatter plots and hyperbolas for data visualization. The construction employs commands such as Line, OrthogonalLine, Point, Intersect, Polygon, Angle, and Segment to build the rectangle and its auxiliary lines, generating angles α, β, γ, δ and segments AD, BC, along with text1 for labeling key information. It is suitable for middle school geometry or introductory calculus, helping students intuitively grasp rectangle properties and the geometric meaning of area computation, while observing how parameter changes affect the figure in a dynamic environment.