A rigid right triangle with side length AB = 5 , AC = 4 AND BC = 3 . If side AB can slide along x and y axes . Find the locus of C 

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,

A rigid right triangle with side length AB = 5, AC = 4, and BC = 3. If side AB can slide along the x and y axes, find the locus of C.

Explore how to find the locus of point C in a rigid right triangle with sides AB = 5, AC = 4, and BC = 3 as it slides along the x and y axes. This problem involves coordinate geometry, the Pythagorean theorem, and the distance formula. Ideal for high school students.

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