## 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(x_{1}, y_{1}) and Q(x_{2}, y_{2}) be any two points on the cartesian plane.

ΔPTQ is right-angled at T.

By Pythagoras Theorem**,**

PQ^{2} = PT^{2} + QT^{2}=>(x_{2} – x_{1})^{2 }+ (y_{2} – y_{1})^{2}**PQ = √[x**_{2} – x_{1})^{2 }+ (y_{2} – y_{1})^{2}]

#### Sections formula-

If the point P(x, y) divides the line segment joining A(x_{1}, y_{1}) and B(x_{2}, y_{2}) 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(x_{1}, y_{1}) and B(x_{2}, y_{2}) is given by: **p(x, y)=(x**_{1}+x_{2}/2 , y_{1}+y_{2}/2)

#### points of trisection-

To find the points of trisection P and Q which divides the line segment joining A(x_{1}, y_{1}) and B(x_{2}, y_{2}) into three equal parts:

#### Centroid of a triangle-

If A(x_{1}, y_{1}), B(x_{2}, y_{2}) and C(x_{3}, y_{3}) 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(x_{1}, y_{1}),B(x_{2}, y_{2}) and C(x_{3}, y_{3}) are the vertices of a Δ ABC, then its area is given by: **A = (1/2)[x**_{1}(y_{2} − y_{3}) + x_{2}(y_{3} − y_{1}) + x_{3}(y_{1} − y_{2})]

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.