Square Root Construction: Geometric Method via Semicircle and Right Triangle

This example demonstrates how to construct the square root of any positive real number geometrically, using a semicircle and right triangle properties to translate arithmetic operations into visual relationships. Core mathematical concepts include square roots, Thales' theorem (angle in a semicircle), the projection theorem in right triangles, and representation of real numbers on the number line. Construction highlights: using GeoGebra commands such as Point, Intersect, Semicircle, Line, OrthogonalLine, Segment, and Polygon, one first builds a semicircle with a point on its diameter, then draws a perpendicular to form a right triangle, and finally determines the square root length via similar triangles or projection relations. This is suitable for upper middle school to high school students and teachers exploring the connection between geometric construction and algebra, and for investigating exact positions of irrational numbers on the number line, deepening understanding of real numbers and geometric intuition.