**Rectangular Cartesian Co-ordinates Geometry**

Co-ordinates Geometry is a system of geometry where the position of points on the plane is described using an ordered pair of numbers. Cartesian coordinates provide a method of rendering graphs and indicating the positions of points on a two-dimensional (2D) surface or in three-dimensional (3D) space. The two axes of two-dimensional Cartesian coordinates, conventionally denoted the x- and y-axes are chosen to be linear and mutually perpendicular.

(i) If the pole and initial line of the polar system coincides respectively with the origin and positive x-axis of the Cartesian system and (x, y), (r, θ) be the Cartesian and polar co-ordinates respectively of a point P on the plane then,

x = r cos θ, y = r sin θ

and r = √(x^{2} + y^{2}), θ = tan^{-1}(y/x).

(ii) The distance between two given points P (x_{1}, y_{1}) and Q (x_{2}, y_{2}) is PQ = √{(x_{2} – x_{1})^{2} + (y_{2} – y_{1})^{2}}.

(iii) Let P (x_{1}, y_{1}) and Q (x_{2}, y_{2}) be two given points.

(a) If the point R divides the line-segment PQ internally in the ratio m : n, then the co-ordinates of R, are {(mx_{2} + nx_{1})/(m + n) , (my_{2} + ny_{1})/(m + n)}.

(b) If the point R divides the line-segment PQ externally in the ratio m : n, then the co-ordinates of R are {(mx_{2} – nx_{1})/(m – n), (my_{2} – ny_{1})/(m – n)}.

(c) If R is the mid-point of the line-segment PQ, then the co-ordinates of R are {(x_{1} + x_{2})/2, (y_{1} + y_{2})/2}.

(iv) The co-ordinates of the centroid of the triangle formed by joining the points (x_{1}, y_{1}) , (x_{2}, y_{2}) and (x_{3}, y_{3}) are ({x_{1} + x_{2} + x_{3}}/3 , {y_{1} + y_{2} + y_{3}}/3

(v) The area of a triangle formed by joining the points (x_{1}, y_{1}), (x_{2}, y_{2}) and (x_{3}, y_{3}) is

½ | y_{1} (x_{2} – x_{3}) + y_{2} (x_{3} – x_{1}) + y_{3} (x_{1} – x_{2}) | sq. units

or, ½ | x_{1} (y_{2} – y_{3}) + x_{2} (y_{3} – y_{1}) + x_{3} (y_{1} – y_{2}) | sq. units.

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