Introduction: Co-ordinate Geometry

Points on cartesian plane-

The coordinates are a pair of numbers that are used to locate points on a plane. x-coordinate is the distance between two points on the y-axis. y-coordinate is the distance between two points on the x-axis.

Representation of (x,y) on the cartesian plane

Distance formula-

The distance between two points that are on the same axis (x-axis or y-axis), is given by the difference between their ordinates if they are on the y-axis, else by the difference between their abscissa if they are on the x-axis.

Distance AB = 6 – (-2) = 8 units
Distance CD = 4 – (-8) = 12 units

Distance formula between two points using Pythagoras theorem-

Let P(x₁, y₁) and Q(x₂, y₂) be any two points on the cartesian plane.
ΔPTQ is right-angled at T.
By Pythagoras Theorem,

PQ² = PT² + QT²=>(x₂ – x₁)² + (y₂ – y₁)²PQ = √[x₂ – x₁)² + (y₂ – y₁)²]

Sections formula-

If the point P(x, y) divides the line segment joining A(x₁, y₁) and B(x₂, y₂) internally in the ratio m:n, then, the coordinates of P are given by the section formula as:

Midpoint-

The midpoint of any line segment divides it in the ratio 1 : 1.The coordinates of the midpoint(P) of line segment joining A(x₁, y₁) and B(x₂, y₂) is given by: p(x, y)=(x₁+x₂/2 , y₁+y₂/2)

points of trisection-

To find the points of trisection P and Q which divides the line segment joining A(x₁, y₁) and B(x₂, y₂) into three equal parts:

Centroid of a triangle-

If A(x₁, y₁), B(x₂, y₂) and C(x₃, y₃) are the vertices of a ΔABC, then the coordinates of its centroid(P) is given by:

Area of a triangle given its vertices-

If A(x₁, y₁),B(x₂, y₂) and C(x₃, y₃) are the vertices of a Δ ABC, then its area is given by: A = (1/2)[x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)]
Where A is the area of the Δ ABC.

Co-linearity condition-
 Three or more points that lie on a same straight line are called collinear points. If three points A, B and C are collinear and B lies between A and C, then:

AB + BC = AC. AB, BC, and AC can be calculated using the distance formula.
The ratio in which B divides AC, calculated using section formula for both the x and y coordinates separately will be equal.

Area of a triangle formed by three collinear points is zero.