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

Options

A. (a) x = y
B. (b) y = z
C. (c) z = x
D. (d) x = y = z

Answer

To determine the conditions under which the points A(x, y), B(y, z), and C(z, x) represent the vertices of a right-angled triangle, we will analyze the slopes of the lines formed by these points.

### Step 1: Identify the points and their coordinates

The points are given as:
- A = (x, y)
- B = (y, z)
- C = (z, x)

### Step 2: Determine the slopes of the lines

1. **Slope of line AB**:
\[ \text{slope of } AB = \frac{z - y}{y - x} \]

2. **Slope of line BC**:
\[ \text{slope of } BC = \frac{x - z}{z - y} \]

### Step 3: Condition for perpendicularity

For the triangle to be a right triangle, the product of the slopes of AB and BC must equal -1 (since perpendicular lines have slopes that multiply to -1):
\[ \left(\frac{z - y}{y - x}\right) \cdot \left(\frac{x - z}{z - y}\right) = -1 \]

### Step 4: Simplify the equation

Cross-multiplying gives:
\[ (z - y)(x - z) = - (y - x)(z - y) \]
Cancelling \(z - y\) (assuming \(z \neq y\)):
\[ x - z = - (y - x) \]
This simplifies to:
\[ x - z = -y + x \]
Thus, we have:
\[ x - z = x - y \]
This leads to:
\[ z = y \]

### Step 5: Analyze other cases

Now, we will check the other cases where the right angle could be at points A or C.

**Case 2: Right angle at A**

1. **Slope of AC**:
\[ \text{slope of } AC = \frac{y - x}{x - z} \]

2. **Slope of AB**:
\[ \text{slope of } AB = \frac{z - y}{y - x} \]

Setting the product of slopes equal to -1:
\[ \left(\frac{y - x}{x - z}\right) \cdot \left(\frac{z - y}{y - x}\right) = -1 \]
Cancelling \(y - x\) (assuming \(y \neq x\)):
\[ (z - y)(y - x) = - (x - z)(y - x) \]
This simplifies to:
\[ z - y = - (x - z) \]
Thus, we have:
\[ y = x \]

**Case 3: Right angle at C**

1. **Slope of BC**:
\[ \text{slope of } BC = \frac{x - z}{z - y} \]

2. **Slope of AC**:
\[ \text{slope of } AC = \frac{y - x}{x - z} \]

Setting the product of slopes equal to -1:
\[ \left(\frac{x - z}{z - y}\right) \cdot \left(\frac{y - x}{x - z}\right) = -1 \]
Cancelling \(x - z\) (assuming \(x \neq z\)):
\[ (y - x)(x - z) = - (z - y)(x - z) \]
This simplifies to:
\[ y - x = z - y \]
Thus, we have:
\[ x = z \]

### Conclusion

From the three cases, we have the following conditions:
1. \(z = y\) (right angle at B)
2. \(y = x\) (right angle at A)
3. \(x = z\) (right angle at C)

Therefore, the points A, B, and C represent the vertices of a right-angled triangle if any of the following conditions hold:
- \(x = z\)
- \(y = x\)
- \(z = y\)