Five-Polygon and Vector Demo

Create a “five-polygon and vector demo” figure for high-school math. First use random parameters and conditional logic to determine a set of five vertex label placements, then shuffle those labels and select one by index for consistent annotation. Build a pentagon in the plane using the resulting vertex positions, marking its vertices and sides. Choose a directed segment between two relevant points as a vector and depict its direction using the polygon’s geometry, with emphasis on trigonometric relations (angles/coordinates) that support vector representation. Add clear mathematical text labels indicating items such as correctness and the number of problems, placed as part of the demonstration layout.