Classic Square Root Construction via Semicircle and Perpendicular

① This example demonstrates the classical geometric construction of square roots on a number line using a semicircle and perpendicular lines, visually linking algebra and geometry. ② Core mathematical concepts include square roots, similar right triangles, the theorem that an angle inscribed in a semicircle is a right angle, and the geometric representation of irrational numbers. ③ Construction details: using GeoGebra commands such as Semicircle, Line, OrthogonalLine, Intersect, Segment, Square, and Sequence, the applet builds a semicircle, drops perpendiculars to intersect the circle, and constructs successive squares to display a sequence of square roots. ④ Suitable for middle school to high school students, this interactive example helps learners grasp the geometric meaning of square roots, explore the Pythagorean theorem, and understand dynamic construction workflows in GeoGebra, making it ideal for classroom demonstrations or self-guided exploration.