Points on a Circle at Integer Distances from the Origin

This applet explores how many points on the given circle (x-4)^2+(y-3)^2=4 have an integer distance to the origin. By calculating the distance from the center to the origin and considering the radius, we can determine the range of distances from the origin to points on the circle, and then identify the integer values within this range. Geometric constructions are used to find the corresponding intersection points. This applet combines coordinate geometry with geometric construction to intuitively display the distribution of points under distance constraints. Students can drag the circle or adjust parameters to observe how the number of points with integer distances changes dynamically, deepening their understanding of the relationship between algebraic equations and geometric figures.