Ratios Within Right Triangles

① This example dynamically demonstrates the proportional relationships among segments within a right triangle, such as the altitude to the hypotenuse, projections of the legs, and the legs themselves, revealing constant ratios derived from similar triangles. ② Core mathematical concepts include similarity of right triangles, proportional segments, trigonometric ratios (sine, cosine, tangent), and the geometric interpretation of the Pythagorean theorem. ③ Construction highlights: using GeoGebra commands Polygon to create the right triangle, Angle to mark the right angle, Line to draw the altitude to the hypotenuse, Intersect to find the foot of the altitude, and multiple text objects (text1 to text10) to display segment lengths and ratios dynamically; an auxiliary triangle (poly2) and angles α and β illustrate equal corresponding angles. ④ Suitable for upper middle school to high school students, this interactive tool aids in teaching geometric proportions, similar triangles, and trigonometry, and serves as a reference for teachers creating dynamic lessons.