# Cartesian Coordinates

Two perpendicular real axes in the plane define a (rectangular planar) Cartesian coordinate system.

Their common point is taken to be the origin (for both of them) and the scale of the two axis are commonly equal. Usually, the horizontal axis is called x-axis, the vertical one is called y-axis.

# Distance between two points

**(i) Distance Formula:**

Let coordinates of A â‰¡ () and coordinates of B â‰¡ ()

The distance between A and B.

AB =

AB =

The distance of point P () from origin O (0, 0):

OP =

**(ii)** **Section formula:**

Consider two points A () and B ()

i. Let a points P (x , y) divides the segment AB in the ratio of m : n â‡¨

The coordinates of point P (x, y) are:

x

y

ii. Let a point Q (x, y) divides the segment AB externally in the ratio of m : n â‡¨

The coordinates of point Q (x, y) are :

x

y

Mathematically the case of external division can be taken as a case of internal division in the ration m : -n.

**(iii) Mid-point formula:**

If P (x, y) is the mid-point of AB, then m : n :: 1 : 1, then the coordinates of point P are :

P (x, y)

**(iv) ** **Area of Triangle**

Consider a triangle ABC with vertices A (), B (), C ().

The area of triangle ABC is denoted by and is given as:

Area = =