The points `A(x,y), B(y, z)` and C(z,x) represents the vertices of a right angled triangle, if

Options

A. x = y
B. y = z
C. z = x
D. x = y = z

Answer

To solve the problem of determining the conditions under which the points \( A(x,y) \), \( B(y,z) \), and \( C(z,x) \) form a right-angled triangle, we will analyze three cases: when the right angle is at point A, point B, and point C respectively.

### Step-by-Step Solution

**Case 1: Right Angle at A (Point A is the right angle)**

1. **Identify the slopes of sides BA and CA:**
- Slope of BA:
\[ m_{BA} = \frac{z - y}{y - x} \]
- Slope of CA:
\[ m_{CA} = \frac{x - y}{z - x} \]

2. **Set the product of the slopes equal to -1 (condition for perpendicularity):**
\[ m_{BA} \cdot m_{CA} = -1 \]
Substituting the slopes:
\[ \frac{z - y}{y - x} \cdot \frac{x - y}{z - x} = -1 \]

3. **Cross-multiply and simplify:**
\[ (z - y)(x - y) = -(y - x)(z - x) \]
Expanding both sides:
\[ zx - zy - yx + y^2 = -yz + yx + xz - x^2 \]
Rearranging gives:
\[ zx - zy + x^2 - yz + y^2 = 0 \]
After simplification, we find:
\[ z - y = z - x \implies x = y \]

**Case 2: Right Angle at B (Point B is the right angle)**

1. **Identify the slopes of sides AB and CB:**
- Slope of AB:
\[ m_{AB} = \frac{z - y}{z - x} \]
- Slope of CB:
\[ m_{CB} = \frac{x - z}{z - y} \]

2. **Set the product of the slopes equal to -1:**
\[ m_{AB} \cdot m_{CB} = -1 \]
Substituting the slopes:
\[ \frac{z - y}{z - x} \cdot \frac{x - z}{z - y} = -1 \]

3. **Cross-multiply and simplify:**
\[ (z - y)(x - z) = -(z - x)(z - y) \]
This simplifies to:
\[ z - y = -y \implies y = z \]

**Case 3: Right Angle at C (Point C is the right angle)**

1. **Identify the slopes of sides AC and BC:**
- Slope of AC:
\[ m_{AC} = \frac{y - x}{x - z} \]
- Slope of BC:
\[ m_{BC} = \frac{z - x}{z - y} \]

2. **Set the product of the slopes equal to -1:**
\[ m_{AC} \cdot m_{BC} = -1 \]
Substituting the slopes:
\[ \frac{y - x}{x - z} \cdot \frac{z - x}{z - y} = -1 \]

3. **Cross-multiply and simplify:**
\[ (y - x)(z - x) = -(x - z)(z - y) \]
This simplifies to:
\[ y - x = -z \implies x = z \]

### Conclusion

From the three cases, we have derived the following conditions:
- Right angle at A: \( x = y \)
- Right angle at B: \( y = z \)
- Right angle at C: \( x = z \)

Thus, the points \( A(x,y) \), \( B(y,z) \), and \( C(z,x) \) can form a right-angled triangle under the conditions derived above.