If `z in C`, then least value of the expression `|z| + |1-z| + |z-2|` is

Options

A. 4
B. `3//2`
C. 2
D. cannot be determined

Answer

To find the least value of the expression \( |z| + |1 - z| + |z - 2| \), we can interpret it geometrically in the complex plane.

### Step-by-Step Solution:

1. **Understanding the Expression**:
The expression \( |z| + |1 - z| + |z - 2| \) represents the sum of distances from the point \( z \) to the points \( 0 \), \( 1 \), and \( 2 \) in the complex plane.

2. **Geometric Interpretation**:
- \( |z| \) is the distance from \( z \) to the origin \( 0 \).
- \( |1 - z| \) is the distance from \( z \) to the point \( 1 \) (which is \( (1, 0) \) in the complex plane).
- \( |z - 2| \) is the distance from \( z \) to the point \( 2 \) (which is \( (2, 0) \) in the complex plane).

3. **Finding the Minimum Distance**:
To minimize the total distance, we can visualize the points \( 0 \), \( 1 \), and \( 2 \) on the real axis. The minimum distance occurs when \( z \) lies on the line segment connecting these points.

4. **Using the Triangle Inequality**:
By the triangle inequality, the sum of the distances from a point to two other points is minimized when the point lies on the line segment connecting those two points. Thus, we can consider the points \( 0 \), \( 1 \), and \( 2 \):
- The distance from \( 0 \) to \( 1 \) is \( 1 \).
- The distance from \( 1 \) to \( 2 \) is \( 1 \).
- Therefore, the minimum distance occurs when \( z \) is at \( 1 \).

5. **Calculating the Minimum Value**:
At \( z = 1 \):
- \( |z| = |1| = 1 \)
- \( |1 - z| = |1 - 1| = 0 \)
- \( |z - 2| = |1 - 2| = 1 \)

Thus, the total distance is:
\[ |z| + |1 - z| + |z - 2| = 1 + 0 + 1 = 2 \]

6. **Conclusion**:
The least value of the expression \( |z| + |1 - z| + |z - 2| \) is \( 2 \).