An isosceles right traingle ABC with BC = AC slides on the axes with A and B on x and y axes respectively. The locus of C is …………… .

Answer

To find the locus of point C in the isosceles right triangle ABC, where A is on the x-axis, B is on the y-axis, and BC = AC, we can follow these steps:

### Step 1: Set Up the Coordinates
Let the coordinates of points A and B be:
- A = (h, 0) on the x-axis
- B = (0, k) on the y-axis

Since C is the right angle vertex of the triangle, its coordinates can be represented as C = (h, k).

### Step 2: Use the Isosceles Right Triangle Property
In an isosceles right triangle, the lengths of the two legs (BC and AC) are equal. Therefore, we have:
- Length of AC = Length of BC

Using the distance formula, we can express these lengths:
- AC = √[(h - 0)² + (k - 0)²] = √(h² + k²)
- BC = √[(h - h)² + (k - 0)²] = √(k²)

Since AC = BC, we can set these equal:
\[ \sqrt{h^2 + k^2} = \sqrt{k^2} \]

### Step 3: Square Both Sides
Squaring both sides to eliminate the square roots gives:
\[ h^2 + k^2 = k^2 \]

### Step 4: Simplify the Equation
Subtract \( k^2 \) from both sides:
\[ h^2 = 0 \]

### Step 5: Interpret the Result
This implies that \( h = 0 \). However, we need to consider the movement of point C as A and B slide along the axes.

### Step 6: Generalize the Locus
Since A can be at any point (h, 0) and B can be at any point (0, k), we can express the relationship between x and y coordinates of point C:
- From the triangle properties, we know that:
- C lies on the line \( y = x \) (since both legs are equal and the triangle is isosceles).
- C also lies on the line \( y = -x \) (the reflection across the origin).

### Conclusion
Thus, the locus of point C as the triangle slides along the axes is given by the equations:
\[ y = x \quad \text{and} \quad y = -x \]