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 (x_{1}, y_{1}) and (x_{2}, y_{2}) 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 m_{1} and m_{1} 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(x_{1},y_{1}) , B(x_{2},y_{2}) and C(x_{3},y_{3}) 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 |