In 2D coordinate geometry we require at least 2 numbers to represent each and every point in 2D plane.

This can be done in a number of ways. Two of the most common ways are Cartesian representation and polar representation. In current article we would be working with Cartesian system which is named after mathematician *Rene* *Descartes’.*

In this system we have two lines which are perpendicular to each other. We conventionally call the horizontal line **x-axis** and the vertical line** y-axis** refer fig.

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**How to represent a point in 2 Dimensional Coordinate Geometry?**

Let’s take an example.

Problem Statement: How to represent a point which is at a distance 3 units from y-axis and 2 units from x-axis? refer fig.

We see that to get a point 3 units away from y-axis we have to move 3 units along x-axis and to get a point 2 units away from x-axis we have to move 2 units along y-axis. 3 here is called the** x-coordinate** or** abscissa** of the point and 2 is called the **y-coordinate** or **ordinate** of the point. The point on whole is represented as (3,2).

In General, any point in 2D space can be represented as (x, y) where x is the abscissa (or x co-ordinate) and y is the ordinate (or y co-ordinate). x is the distance from y-axis and y is the distance from x-axis. The point (0, 0) is called **the origin.**

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## How to find distance between two points in 2D Coordinate Geometry?

Distance formula is used to find the shortest distance between two points in 2 dimensional plane.

In the adjacent figure we have to find the distance between the points A (x_{1}, y_{1}) and C (x_{2}, y_{2}). To find the distance we follow these steps:

- Construct a right angle triangle with 90
^{0}at B. - Notice that co-ordinates of B is (x
_{2}, y_{1}) - AB = x
_{2}– x_{1} - BC = y
_{2}– y_{1} - The distance between A and C can be determined by Pythagoras theorem:

** d=√((x _{2} – x_{1} )^{2} +(y_{2} – y_{1})^{2})**

This is called the distance formula to determine the distance between two known points (x_{1}, y_{1}) & (x_{2}, y_{2})

## Section Formula:

Section formula is used to determine the co-ordinates of a point that divides a line segment in a given ratio.

In the adjacent figure there are two given points A (x_{1}, y_{1}) and B (x_{2}, y_{2}). We have to find out the coordinates of the point C such that C divides the line segment AB in the ratio m_{1}: m_{2}.

- Steps to find the coordinates of C in terms of x
_{1}, y_{1}, x_{2}, y_{2}, m_{1}and m_{2}. - Assume coordinates of C to be (x, y)
- Given AC : CB = m
_{1}: m_{2} - Construct two right angled triangles ADB and CEB as shown in the figure.
- Notice: coordinates of E = (x
_{2}, y) and coordinates of D = (x_{2}, y_{1}) - We have AD = x
_{2}– x_{1}, CE = x_{2}– x, BD = y_{2}– y_{1}and BE = y_{2}– y. - In ∆ADB and ∆CEB
- ∠DAB= ∠ECB & ∠ADB= ∠CEB= 90
^{0} - Thus ∆ADB ~ ∆CEB

Hence

AD/CE=AC/CB

We know AB = AC + CB

AD/CE=(AC+CB)/CB=(AC/CB)+1= m_{1}/m_{2} +1

(x_{2}-x_{1})/(x_{2}-x)= (m_{1}/m_{2} ) +1

From here we can find out that**x=(m _{1} x_{2} + m_{2} x_{1})/(m_{1} + m_{2} )**

**Similarly y=(m _{1} y_{2} + m_{2} y_{1})/(m_{1} + m_{2})**

**This is called the section formula.**

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