The complex numbers z1, z2, z3 and z4 represent the vertices...

The complex numbers $z_{1}, z_{2}, z_{3}$ and $z_{4}$ represent the vertices of a parallelogram taken in order if and only if (a) $\mathrm{z}_{1}+\mathrm{z}_{3}=\mathrm{z}_{2}+\mathrm{z}_{4}$ (b) $\mathrm{z}_{1}+\mathrm{z}_{2}+\mathrm{z}_{3}=\mathrm{z}_{4}$ (c) $\mathrm{z}_{1}+\mathrm{z}_{2}=\mathrm{z}_{3}+\mathrm{z}_{4}$ (d) $\mathrm{z}_{1}-\mathrm{z}_{2}=\mathrm{z}_{3}-\mathrm{z}_{4}$

The complex numbers $z_{1}, z_{2}, z_{3}$ and $z_{4}$ represent the vertices of a parallelogram taken in order if and only if
(a) $\mathrm{z}_{1}+\mathrm{z}_{3}=\mathrm{z}_{2}+\mathrm{z}_{4}$
(b) $\mathrm{z}_{1}+\mathrm{z}_{2}+\mathrm{z}_{3}=\mathrm{z}_{4}$
(c) $\mathrm{z}_{1}+\mathrm{z}_{2}=\mathrm{z}_{3}+\mathrm{z}_{4}$
(d) $\mathrm{z}_{1}-\mathrm{z}_{2}=\mathrm{z}_{3}-\mathrm{z}_{4}$

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Key Concepts

Complex Numbers as Points in the Plane

Complex numbers can be interpreted as points in the two-dimensional plane, where the real part represents the x-coordinate and the imaginary part represents the y-coordinate. This allows problems in geometry to be approached using algebraic manipulations of complex numbers, enabling a seamless blend of algebra and geometry.

Parallelogram Properties: Diagonals Bisect Each Other

One of the defining properties of a parallelogram is that its diagonals intersect at a point that divides them into equal parts. This midpoint condition is crucial because when the vertices are represented as points (or complex numbers), the equality of the sums of opposite vertices implies that their midpoints coincide, thus confirming the parallelogram structure.

Vector Addition in the Complex Plane

In the complex plane, addition of complex numbers corresponds to vector addition, which is fundamental in analyzing geometric figures such as parallelograms. By treating the sides or vertices of a shape as vectors, one can employ algebraic methods to verify geometric properties, like the condition for a quadrilateral to be a parallelogram.

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