Changes Over Rational Intervals: Equilateral Triangles and Mode

This example demonstrates how quantities derived from an equilateral-triangle configuration change when the underlying parameter is sampled over rational intervals, linking geometric variation to a statistical viewpoint. The key ideas are rational numbers and rational-interval sampling, equilateral-triangle geometry constraints, parameter-driven construction of measurable quantities (e.g., lengths/areas depending on the setup), and the statistical concept of the mode (most frequently occurring value) to describe the distribution of outcomes. Construction is organized around multiple point objects (A1–A7 and B1, B2, B4, B6, B7) that serve as vertices and reference points for different parameter positions; as the rational-interval parameter changes, these points update the triangle’s configuration and the resulting values used for statistical interpretation. Suitable for advanced secondary students and early undergraduate lessons on parameter dependence, geometric construction, and introductory probability & statistics.