If `z_(1)=1+2i, z_(2)=2+3i, z_(3)=3+4i`, then `z_(1),z_(2)` and `z_(3)` represent the vertices of a/an.

Options

A. equilateral triangle
B. right angled triangle
C. isosceles triangle
D. none of these

Explanation

To determine the relationship between the complex numbers \( z_1 = 1 + 2i \), \( z_2 = 2 + 3i \), and \( z_3 = 3 + 4i \), we will check if these points are collinear, which means they lie on a straight line.

### Step-by-Step Solution:

1. **Identify the Complex Numbers**:
- Let \( z_1 = 1 + 2i \)
- Let \( z_2 = 2 + 3i \)
- Let \( z_3 = 3 + 4i \)

2. **Calculate the Midpoint of \( z_1 \) and \( z_3 \)**:
- The formula for the midpoint \( M \) of two points \( z_a \) and \( z_b \) in the complex plane is given by:
\[ M = \frac{z_a + z_b}{2} \]
- Here, we calculate the midpoint of \( z_1 \) and \( z_3 \):
\[ M = \frac{z_1 + z_3}{2} = \frac{(1 + 2i) + (3 + 4i)}{2} = \frac{(1 + 3) + (2 + 4)i}{2} = \frac{4 + 6i}{2} = 2 + 3i \]

3. **Compare the Midpoint with \( z_2 \)**:
- We found that the midpoint \( M = 2 + 3i \), which is exactly equal to \( z_2 \).
- This means that \( z_2 \) lies on the line segment connecting \( z_1 \) and \( z_3 \).

4. **Conclusion**:
- Since \( z_2 \) is the midpoint of \( z_1 \) and \( z_3 \), the points \( z_1 \), \( z_2 \), and \( z_3 \) are collinear.
- Therefore, they do not form a triangle.

### Final Answer:
The points \( z_1, z_2, \) and \( z_3 \) represent the vertices of **none of these** (they are collinear).