We are given a rigid right triangle $\triangle ABC$ with side lengths $AB = 5$, $AC = 4$, and $BC = 3$. The triangle can slide along the $x$-axis and $y$-axis, meaning that point $A$ moves along the axes, and we are asked to find the locus of point $C$.

Step-by-Step Breakdown:

1. Coordinate Setup:

   • Let the coordinates of point $A$ be $(x_1, y_1)$.
   • Since $AB = 5$, and point $B$ is constrained to slide along the axes, point $B$ is either on the $x$-axis or $y$-axis. Thus, we assume that the position of $B$ is at the origin, i.e., $B(0, 0)$.

2. Triangle Relations: Since $AB = 5$, the distance between point $A(x_1, y_1)$ and point $B(0, 0)$ must satisfy the equation of a circle centered at $B$ with radius 5: $x_1^2 + y_1^2 = 25.$

3. Finding the Locus of $C$: The distance from $A$ to $C$ is given as $AC = 4$, and the distance from $B$ to $C$ is $BC = 3$. Since $A$ and $B$ are on fixed axes and $C$ forms a right triangle, we apply the Pythagorean theorem. This results in the coordinates of $C$ depending on the position of $A$ and its movement.

However, to fully determine the mathematical equation for the locus,