Interactive Construction of 30°-60°-90° and 45°-45°-90° Right Triangles

This interactive example demonstrates the geometric properties of the special right triangles 30°-60°-90° and 45°-45°-90°, focusing on their angle structure and the resulting side-length relationships. The key math ideas include right-triangle angle constraints, the basis for special triangle ratios at 30° and 45°, and how these relationships align with the Pythagorean theorem. Construction workflow: it creates points and segments using PointIn, Segment, Ray, and Line; marks and controls the critical angles with Angle (labeled angles such as α, β, γ); builds perpendicular structure via OrthogonalLine; generates corresponding triangle configurations through transformations like Rotate, Mirror, and Translate (yielding separate instances such as tri30 and tri45); computes necessary intersections using Intersect; and forms triangle regions with Polygon. Objects and annotations such as the main triangle instances (tri30, tri45), constructed points (A, B, C, D, E, etc.), and text labels (text30, text45) support clear visualization. Suitable for middle/high school geometry lessons, especially for understanding special right triangles and solving related problems using visual reasoning.