Cartesian Coordinates

For locating the position of a point on a plane, we require a pair of coordinate axes.

X and Y Axes

The left-right (horizontal) direction is commonly called X or The distance of a point from the y-axis is called its x-coordinate, or abscissa.

The up-down (vertical) direction is commonly called Y or The distance of a point from the x-axis is called its y-coordinate, or ordinate

The coordinates of a point on the x-axis are of the form (x, 0), and of a point on the y-axis are of the form (0, y).

The point where x-axis and y-axis intersect is called the origin. Its coordinates are (0,0).

The ordinates of all points on a horizontal line which is parallel to x-axis are equal i.e. y = constant = 2.

The abscissa of all points on a vertical line which is a line parallel to y-axis are equal i.e. x = constant = 4.

Four Quadrants of Coordinate Plane

The rectangular axes X’OX and Y’OY divide the plane into four quadrants as below :

The coordinates of the points in the four quadrants will have sign according to the below table:

Quadrant x-coordinate

(abscissa)

y-coordinate

(ordinate)

Ist Quadrant + +
IInd quadrant +
IIIrd quadrant
IVth quadrant +

 

DISTANCE FORMULA

The distance between two points (x1, y1) and (x2, y2) in a rectangular coordinate system is equal to

The distance of a point (x, y) from origin is

SECTION FORMULA

A point P(x,y) which divide the line segment AB in the ratio m1 and m1 is given by

The mid point P is given by

Remark : To remember the section formula, the diagram given below is helpful:

 

Area of Triangle ABC

Area of triangle ABC of coordinates A(x1,y1) , B(x2,y2) and C(x3,y3) is given by

For point A,B and C to be collinear, The value of A should be zero, i.e. A=0.

How to solve general Problems of Area in Coordinate Geometry

Area of Triangle Three vertices will be given, you can calculate the area directly using formula
Area of Square Two vertices will be given, we can calculate either side or diagonal depending on vertices given and apply the square area formula
Area Of rhombus Given: all the vertices coordinates
Two ways
1) Divide the rhombus into two triangle. Calculate the area of both the triangle and sum it
2) Calculate the diagonal and apply the Area formula
Area of parallelogram Three vertices are sufficient to find the area of parallelogram
Calculate the area of the triangle formed by the three vertices and double it to calculate the area of parallelogram
Area of quadrilateral Given: all the vertices coordinates
Divide into two triangle. Calculate the area separately and sum it

 

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