if `A(z_1),B(z_2),C(z_3),D(z_4)` lies on |z|=4 (taken in order) , where `z_1+z_2+z_3+z_4=0` then :

Options

A. Max. area of quadrilateral ABCD=32
B. Max. area of quadrilateral ABCD=16
C. The triangle `DeltaABC` is right angled
D. The quadrilateral ABCD is rectangle

Answer

To solve the problem, we need to analyze the given conditions and derive the necessary conclusions step by step.

### Step-by-Step Solution:

1. **Understanding the Given Information**:
We have four complex numbers \( z_1, z_2, z_3, z_4 \) that lie on the circle defined by \( |z| = 4 \). This means the modulus of each complex number is 4.

2. **Equation of the Circle**:
The equation \( |z| = 4 \) can be expressed in Cartesian coordinates as:
\[ x^2 + y^2 = 16 \]
This represents a circle centered at the origin (0, 0) with a radius of 4.

3. **Sum of the Complex Numbers**:
We are given that:
\[ z_1 + z_2 + z_3 + z_4 = 0 \]
This implies that the centroid of the points \( z_1, z_2, z_3, z_4 \) is at the origin.

4. **Geometric Interpretation**:
Since the points are symmetrically distributed around the origin and lie on the circle, they can be represented as:
\[ z_1 = 4e^{i\theta_1}, \quad z_2 = 4e^{i\theta_2}, \quad z_3 = 4e^{i\theta_3}, \quad z_4 = 4e^{i\theta_4} \]
where \( \theta_1, \theta_2, \theta_3, \theta_4 \) are angles that satisfy the condition of their sum being zero.

5. **Maximum Area of Quadrilateral ABCD**:
To find the maximum area of the quadrilateral formed by these points, we can use the formula for the area of a quadrilateral inscribed in a circle:
\[ \text{Area} = \frac{1}{2} \times d_1 \times d_2 \]
where \( d_1 \) and \( d_2 \) are the lengths of the diagonals.

6. **Finding the Length of the Diagonals**:
The maximum area occurs when the quadrilateral is a square. In this case, the diagonals are equal and can be calculated as:
\[ d_1 = d_2 = 8 \quad (\text{since } 4 + 4 = 8) \]

7. **Calculating the Area**:
The area of the square is:
\[ \text{Area} = \frac{1}{2} \times 8 \times 8 = 32 \]

8. **Right-Angle Triangle**:
Since ABCD is a square, any triangle formed by three of its vertices (like triangle ABC) will also be a right triangle.

9. **Rectangle Confirmation**:
A square is a special case of a rectangle, thus confirming that quadrilateral ABCD is also a rectangle.

### Conclusion:
The maximum area of quadrilateral ABCD is 32, triangle ABC is a right triangle, and quadrilateral ABCD is a rectangle.