Key Concepts and Formulas

• Geometric Representation of Complex Numbers: Complex numbers can be represented as points or vectors in the complex plane. Addition corresponds to vector addition, and multiplication by $i$ corresponds to a 90-degree counter-clockwise rotation.
• Area of a Triangle: The area of a triangle is given by $\frac{1}{2} \times \text{base} \times \text{height}$. For a right-angled triangle, the area is half the product of the lengths of the two perpendicular sides.
• Magnitude of a Complex Number: The magnitude (or modulus) of a complex number $z = a + bi$ is given by $|z| = \sqrt{a^2 + b^2}$. Also, $|iz| = |i| |z| = |z|$.

Step-by-Step Solution

Step 1: Visualize the vertices in the complex plane.

We are given the vertices of a triangle as $A(z)$, $B(iz)$, and $C(z + iz)$. Let's represent these as points in the complex plane. Multiplying $z$ by $i$ rotates the corresponding vector by 90 degrees counterclockwise. Adding $z$ and $iz$ gives us the fourth vertex of a parallelogram formed by the origin, $z$, and $iz$.

Step 2: Determine the side vectors.

To find the area, we'll calculate the vectors representing the sides of the triangle:

• $\vec{AC} = C - A = (z + iz) - z = iz$
• $\vec{BC} = C - B = (z + iz) - iz = z$
• $\vec{AB} = B - A = iz - z = (i-1)z$

Step 3: Calculate the lengths of the sides.

Now, we find the magnitudes of the side vectors:

• $|\vec{AC}| = |iz| = |i| |z| = |z|$
• $|\vec{BC}| = |z|$
• $|\vec{AB}| = |(i-1)z| = |i-1||z| = \sqrt{(-1)^2 + 1^2} |z| = \sqrt{2} |z|$

Step 4: Identify the type of triangle.

We have the side lengths: $AC = |z|$, $BC = |z|$, and $AB = \sqrt{2}|z|$. Since $AC = BC$, the triangle is isosceles. Now, check if it's a right-angled triangle using the Pythagorean theorem: $AC^2 + BC^2 = |z|^2 + |z|^2 = 2|z|^2$ $AB^2 = (\sqrt{2}|z|)^2 = 2|z|^2$ Since $AC^2 + BC^2 = AB^2$, the triangle is right-angled at $C$. Furthermore, since $\vec{AC} = iz$ and $\vec{BC} = z$, $\vec{AC}$ is a 90-degree rotation of $\vec{BC}$, confirming the right angle at $C$.

Step 5: Calculate the area of the triangle.

Since $\triangle ABC$ is a right-angled triangle with the right angle at $C$, the area is given by: Area $= \frac{1}{2} \times AC \times BC = \frac{1}{2} \times |z| \times |z| = \frac{1}{2} |z|^2$

Common Mistakes & Tips

• Incorrectly Applying Area Formulas: Avoid blindly using formulas without understanding the geometry. Visualizing the complex numbers as vectors makes it easier to recognize the right-angled triangle.
• Sign Errors: Be careful with vector subtraction and ensuring you are calculating the vector from one point to another correctly.
• Modulus Properties: Remember that $|iz| = |z|$ and $|cz| = |c||z|$ for any complex number $z$ and scalar $c$.

Summary

By representing the vertices of the triangle as complex numbers and using the geometric interpretation of complex number operations, we determined that the triangle is a right-angled isosceles triangle with legs of length $|z|$. Therefore, the area of the triangle is $\frac{1}{2}|z|^2$.

The final answer is $\boxed{\frac{1}{2}|z|^2}$, which corresponds to option (B).