Question 1210495: Calculate the area of the triangle formed by the complex numbers , , and in the Argand plane, where is a complex number.
Answer by CPhill(2138)   (Show Source):
You can put this solution on YOUR website! The area of the triangle formed by the complex numbers $z_1 = 0$, $z_2 = 1$, and $z_3 = i$ in the Argand plane is **$\frac{1}{2}$ square units**.
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### Step 1: Identify the Coordinates
In the Argand plane, a complex number $z = x + yi$ corresponds to the Cartesian coordinate point $(x, y)$.
1. $z_1 = 0 = 0 + 0i \implies P_1 = (0, 0)$
2. $z_2 = 1 = 1 + 0i \implies P_2 = (1, 0)$
3. $z_3 = i = 0 + 1i \implies P_3 = (0, 1)$
### Step 2: Determine the Type of Triangle
Plotting these points reveals a right-angled triangle:
* The side from $P_1$ to $P_2$ lies along the positive **Real axis** (the $x$-axis) and has a length of **1**.
* The side from $P_1$ to $P_3$ lies along the positive **Imaginary axis** (the $y$-axis) and has a length of **1**.
* Since the Real and Imaginary axes are perpendicular, the angle at the origin ($P_1$) is $90^\circ$.
### Step 3: Calculate the Area
The area of a right-angled triangle is given by the formula:
$$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$$
In this case:
* Base $= \text{length of the side from } (0, 0) \text{ to } (1, 0) = 1$
* Height $= \text{length of the side from } (0, 0) \text{ to } (0, 1) = 1$
$$\text{Area} = \frac{1}{2} \times 1 \times 1 = \frac{1}{2}$$
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### Alternative Method: Using the Determinant Formula
The area of a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ is also given by:
$$\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|$$
Plugging in the coordinates $(0, 0)$, $(1, 0)$, and $(0, 1)$:
$$\text{Area} = \frac{1}{2} \left| 0(0 - 1) + 1(1 - 0) + 0(0 - 0) \right|$$
$$\text{Area} = \frac{1}{2} \left| 0 + 1(1) + 0 \right|$$
$$\text{Area} = \frac{1}{2} \times 1 = \frac{1}{2}$$
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