The area of the triangle on the Arand plane formed by the complex numbers z, iz and z+iz is?

Options

A. `|z|^(2)`
B. `1/2|z|^(2)`
C. `1/4|z|^(2)`
D. `sqrt(3)/4|z|^(2)`

Explanation

To find the area of the triangle formed by the complex numbers \( z \), \( iz \), and \( z + iz \) in the Argand plane, we can follow these steps:

### Step 1: Identify the vertices of the triangle
The vertices of the triangle are given by the complex numbers:
- \( A = z \)
- \( B = iz \)
- \( C = z + iz \)

### Step 2: Express the vertices in terms of coordinates
We can express these complex numbers in terms of their real and imaginary parts:
- Let \( z = x + iy \), where \( x \) and \( y \) are real numbers.
- Then, \( A = (x, y) \)
- For \( B = iz = i(x + iy) = (-y, x) \)
- For \( C = z + iz = z + i z = (x + iy) + (-y + ix) = (x - y, x + y) \)

### Step 3: Find the coordinates of the vertices
Thus, the coordinates of the vertices are:
- \( A = (x, y) \)
- \( B = (-y, x) \)
- \( C = (x - y, x + y) \)

### Step 4: Calculate the area of the triangle
The area \( A \) of a triangle formed by three points \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
Substituting the coordinates of points \( A \), \( B \), and \( C \):
\[ \text{Area} = \frac{1}{2} \left| x(x - (x + y)) + (-y)((x + y) - y) + (x - y)(y - x) \right| \]

### Step 5: Simplify the expression
Calculating the expression:
\[ = \frac{1}{2} \left| x(-y) + (-y)(x) + (x - y)(y - x) \right| \]
\[ = \frac{1}{2} \left| -xy - xy + (x - y)(y - x) \right| \]
\[ = \frac{1}{2} \left| -2xy + (x - y)(y - x) \right| \]
\[ = \frac{1}{2} \left| -2xy + (x^2 - xy - yx + y^2) \right| \]
\[ = \frac{1}{2} \left| x^2 - 2xy + y^2 \right| \]
\[ = \frac{1}{2} \left| (x - y)^2 \right| \]

### Step 6: Relate to \( |z|^2 \)
Since \( |z|^2 = x^2 + y^2 \), we can express the area in terms of \( |z| \):
\[ = \frac{1}{2} |z|^2 \]

### Final Result
Thus, the area of the triangle formed by the complex numbers \( z \), \( iz \), and \( z + iz \) is:
\[ \text{Area} = \frac{1}{2} |z|^2 \]