If the points A(z), B(-z) and C(1-z) are the vertices of an equilateral triangle, then value of z is

Options

A. `1 +- (i sqrt(3))/(2)`
B. `(1)/(2) (1 +- i)`
C. `(1)/(4) (1 +- sqrt(3)i)`
D. `(1)/(3) (1 +- sqrt(3)i)`

Answer

To solve the problem where points A(z), B(-z), and C(1-z) are the vertices of an equilateral triangle, we will follow these steps:

### Step 1: Understand the properties of an equilateral triangle
In an equilateral triangle, all sides are equal in length. Therefore, we need to find the lengths of the sides AB, BC, and CA and set them equal to each other.

### Step 2: Calculate the length of side AB
The length of side AB can be calculated using the distance formula:
\[ AB = |A - B| = |z - (-z)| = |z + z| = |2z| = 2|z| \]

### Step 3: Calculate the length of side BC
Next, we calculate the length of side BC:
\[ BC = |B - C| = |-z - (1 - z)| = |-z - 1 + z| = |-1| = 1 \]

### Step 4: Calculate the length of side CA
Now, we calculate the length of side CA:
\[ CA = |C - A| = |(1 - z) - z| = |1 - z - z| = |1 - 2z| \]

### Step 5: Set the lengths equal
Since all sides of the triangle are equal, we can set the lengths we calculated equal to each other:
\[ AB = BC \quad \text{and} \quad AB = CA \]
This gives us two equations:
1. \(2|z| = 1\)
2. \(2|z| = |1 - 2z|\)

### Step 6: Solve the first equation
From the first equation \(2|z| = 1\):
\[ |z| = \frac{1}{2} \]

### Step 7: Substitute |z| into the second equation
Now we substitute \(|z| = \frac{1}{2}\) into the second equation:
\[ 1 = |1 - 2z| \]
This means \(1 - 2z = 1\) or \(1 - 2z = -1\).

### Step 8: Solve for z
1. From \(1 - 2z = 1\):
\[ -2z = 0 \implies z = 0 \]

2. From \(1 - 2z = -1\):
\[ 1 + 1 = 2z \implies 2 = 2z \implies z = 1 \]

### Step 9: Verify the values of z
We need to check if \(z = 0\) or \(z = 1\) satisfies the condition of forming an equilateral triangle:
- For \(z = 0\):
- A(0), B(0), C(1) do not form a triangle.

- For \(z = 1\):
- A(1), B(-1), C(0) form a triangle, but we need to check if it's equilateral:
- Lengths: AB = 2, BC = 1, CA = 1 (not equal).

### Conclusion
The only valid solution that satisfies the conditions of the problem is:
\[ |z| = \frac{1}{2} \]

### Final Answer
Thus, the value of \(z\) is such that \(|z| = \frac{1}{2}\).

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