The points A,B and C represent the complex numbers `z_(1),z_(2),(1-i)z_(1)+iz_(2)` respectively, on the complex plane (where, `i=sqrt(-1)`). The `/_\ABC`, is

Options

A. isosceles but not right angled
B. right angled but not isosceles
C. isosceles and right angled
D. None of the above

Answer

To solve the problem, we need to analyze the points A, B, and C represented by the complex numbers \( z_1, z_2, (1-i)z_1 + iz_2 \) respectively. We will find the lengths of the sides of triangle ABC and determine its properties.

### Step-by-Step Solution:

1. **Identify the Points**:
- Let \( A = z_1 \)
- Let \( B = z_2 \)
- Let \( C = (1 - i)z_1 + iz_2 \)

2. **Calculate the Lengths of the Sides**:
- **Length \( AB \)**:
\[ AB = |z_1 - z_2| \]

- **Length \( BC \)**:
\[ BC = |C - B| = |(1 - i)z_1 + iz_2 - z_2| = |(1 - i)z_1 + (i - 1)z_2| \]
Simplifying this:
\[ BC = |(1 - i)z_1 - (1 - i)z_2| = |(1 - i)(z_1 - z_2)| = |1 - i| \cdot |z_1 - z_2| = \sqrt{2} |z_1 - z_2| \]

- **Length \( CA \)**:
\[ CA = |C - A| = |(1 - i)z_1 + iz_2 - z_1| = |(-i)z_1 + iz_2| = |i(z_2 - z_1)| = |z_2 - z_1| = |z_1 - z_2| \]

3. **Summarize the Lengths**:
- We have:
\[ AB = |z_1 - z_2| \]
\[ BC = \sqrt{2} |z_1 - z_2| \]
\[ CA = |z_1 - z_2| \]

4. **Analyze the Triangle**:
- From the lengths, we see that \( AB = CA \) and \( BC = \sqrt{2} AB \).
- Since \( AB = CA \), triangle ABC is isosceles.
- To check if it is a right triangle, we can use the Pythagorean theorem:
\[ AB^2 + CA^2 = BC^2 \]
\[ |z_1 - z_2|^2 + |z_1 - z_2|^2 = (\sqrt{2}|z_1 - z_2|)^2 \]
\[ 2|z_1 - z_2|^2 = 2|z_1 - z_2|^2 \]
- This confirms that triangle ABC is also a right triangle.

### Conclusion:
Triangle ABC is an isosceles right triangle.