If the points z, i z,(z-i z) represent the vertices of a triangle in the Argand plane, then, (a)

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If the points z, i z,(z-i z) represent the vertices of a...

If the points $\mathrm{z}, \mathrm{i} z,(\mathrm{z}-\mathrm{i} z)$ represent the vertices of a triangle in the Argand plane, then, (a) area of the triangle $=\frac{1}{2}|\mathrm{z}|^{2}$ (b) orthocenter is at $(1-\mathrm{i}) \mathrm{z}$ (c) circumcentre is at $\frac{1}{2}(1-i) z$ (d) centroid is at the origin

If the points $\mathrm{z}, \mathrm{i} z,(\mathrm{z}-\mathrm{i} z)$ represent the vertices of a triangle in the Argand plane, then,
(a) area of the triangle $=\frac{1}{2}|\mathrm{z}|^{2}$
(b) orthocenter is at $(1-\mathrm{i}) \mathrm{z}$
(c) circumcentre is at $\frac{1}{2}(1-i) z$
(d) centroid is at the origin

Key Concepts

Centroid of a Triangle

The centroid of a triangle, often considered its center of mass, is the point obtained by averaging the coordinates of its vertices. In the complex plane, this is equivalent to taking the arithmetic mean of the complex numbers representing the vertices, which yields a point that balances the triangle.

Circumcenter of a Triangle

The circumcenter is the point equidistant from all three vertices of a triangle, and it serves as the center of the triangle's circumscribed circle. In the context of the complex plane, the circumcenter is determined by the intersection of the perpendicular bisectors of the sides and is an important concept in triangle geometry, often explored in relation to other centers like the centroid and orthocenter.

Orthocenter of a Triangle

The orthocenter of a triangle is the common intersection point of the three altitudes. While its exact geometric construction can be intricate, particularly in the complex plane, it remains a significant triangle center with properties that link it to the vertices and other centers through various geometric and algebraic relationships.

Complex Multiplication and Rotation

Multiplying a complex number by another complex number, particularly by i, results in a rotation about the origin. Specifically, multiplication by i rotates any complex number by 90° counterclockwise in the plane. This property is essential in analyzing geometric configurations created by transformed versions of an initial point.

Complex Numbers in the Argand Plane

In the Argand plane, every complex number represents a point with coordinates corresponding to its real and imaginary parts. This allows the interpretation of algebraic operations on complex numbers as geometric transformations such as translations, rotations, and dilations in the two-dimensional plane.

Triangle Area Calculation in the Complex Plane

The area of a triangle defined by vertices represented as complex numbers can be calculated using formulas based on determinants or cross products, analogous to methods in coordinate geometry. This approach leverages the geometric representation of complex numbers to calculate distances and orientations, leading to the triangle’s area.

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