Crash Course Mathematics, Coordinate Geometry Introduction

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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coordinate geometry : points 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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Coordinate geometry : distance formulaHow 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 (x1, y1) and C (x2, y2). To find the distance we follow these steps:

  • Construct a right angle triangle with 900 at B.
  • Notice that co-ordinates of B is (x2, y1)
  • AB = x2 – x1
  • BC = y2 – y1
  • The distance between A and C can be determined by Pythagoras theorem:

d=√((x2 – x1 )2 +(y2 – y1)2)

This is called the distance formula to determine the distance between two known points (x1, y1) & (x2, y2)

Section Formula:

coordinate geometry : section formulaSection 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 (x1, y1) and B (x2, y2). We have to find out the coordinates of the point C such that C divides the line segment AB in the ratio m1: m2.

  • Steps to find the coordinates of C in terms of x1, y1, x2, y2, m1 and m2.
  • Assume coordinates of C to be (x, y)
  • Given AC : CB = m1 : m2
  • Construct two right angled triangles ADB and CEB as shown in the figure.
  • Notice: coordinates of E = (x2, y) and coordinates of D = (x2, y1)
  • We have AD = x2 – x1, CE = x2 – x, BD = y2 – y1 and BE = y2 – y.
  • In ∆ADB and ∆CEB
  • ∠DAB= ∠ECB & ∠ADB= ∠CEB= 900
  • Thus ∆ADB ~ ∆CEB

Hence
AD/CE=AC/CB

We know AB = AC + CB
AD/CE=(AC+CB)/CB=(AC/CB)+1= m1/m2 +1

(x2-x1)/(x2-x)= (m1/m2 ) +1

From here we can find out that
x=(m1 x2 + m2 x1)/(m1 + m2 )

Similarly y=(m1 y2 + m2 y1)/(m1 + m2)

This is called the section formula.

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