Maximum Area of a Triangle

This GeoGebra applet demonstrates a classic coordinate geometry and optimization problem. Given the origin O and a fixed point P, points A and B lie on the x-axis and y-axis respectively, with the condition that segment PA is perpendicular to PB. Using geometric construction commands like OrthogonalLine and Polygon, it dynamically visualizes how the triangle's area changes as the positions of A and B vary while maintaining the perpendicularity constraint. Learners can drag the movable points to explore the relationship between the positions of A and B and the resulting area of triangle OAB. The applet helps students intuitively discover the geometric conditions under which the triangle's area reaches its maximum value, bridging visual exploration with analytical reasoning. Through this interactive investigation, students develop a deeper understanding of how perpendicularity constraints in coordinate geometry relate to optimization problems, and they practice connecting geometric intuition with algebraic methods for finding extrema.