Complex Scalar Multiplication: Geometric Scaling via Dilation

This GeoGebra example dynamically illustrates the geometric effect of multiplying a complex number by a real scalar, showing how the complex plane undergoes a uniform scaling transformation. The core mathematical concept is scalar multiplication of complex numbers: when a complex number is multiplied by a real number, its modulus scales by the absolute value of that scalar, while its argument remains unchanged (or rotates by π if the scalar is negative). The construction uses the Polygon command to create a triangle with one vertex at the origin, and the Dilate command to scale the triangle's vertices A, B, C about the origin by a factor controlled by a slider, producing the transformed points A', B', C' and the scaled triangle. A slider allows real-time adjustment of the scalar, making the size change visible while the original triangle remains as a reference. This example is suitable for high school or early college students learning complex numbers and geometric transformations, helping them connect algebraic multiplication with visual scaling, and serving as an introductory tool for linear transformations.