Introduction to Coordinate Geometry / Analytical Geometry

What is Coordinate Geometry?

Coordinate geometry is one of the most important and exciting mathematical disciplines. In particular, it is the backbone of a mathematical interaction at school. Provides links between algebra and geometry with line graphs and curves.

Introduction

The Cartesian coordinate invented in the 17th century by Rene Descartes (Latinized name as Cartesius) revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes (such as lines and curves) can be described by equations.

Distance Formula

Explain and define Coordinate Geometry 

Coordinate geometry is one of the most important and exciting mathematical disciplines. In particular, it is the backbone of a mathematical interaction at school. Provides links between algebra and geometry with line graphs and curves.

The algebraic study of geometry with the help of a coordinate system is called co-ordinate geometry/analytical geometry.

This enables geometrical problems to be solved algebraically and provides geometric insights into algebra. It is a part of geometry in which ordered pairs of numbers are used to describe the position of a point on a plane. Here, the concept of coordinate geometry (also known as Cartesian geometry) and its formulas and their derivations will be explained.

Derive distance formula to calculate the distance between two given points in the Cartesian plane

Statement 

The distance between any two points `P_1\left(x_1,y_1\right)` and `P_2\left(x_2,y_2\right)` is denoted as |`\overline{P_1P_2}`| and is define as: 

|`\overline{P_1P_2}`|`=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}`

Let `P_1\left(x_1,y_1\right)` and `P_2\left(x_2,y_2\right)` be the any two points in plane.

From `P_1` and `P_2` draw perpendicular `\overline{P_1A}` and `\overline{P_2A}` on x axis, also draw a `\overline{P_1C}` parallel to x-axis.

|`\overline{P_1C}`| `=` |`\overline{AB}`| `=` |`\overline{OB}`| `-` |`\overline{OA}`| `=` |`\overline{x_2-x_1}`| 

and 

|`\overline{P_2C}`| `=` |`\overline{P_2B}`| `-` |`\overline{AB}`| `=` |`\overline{y_2-y_1}`|

Consider right angled `\triangle P_1CP_2`

Applying Pythagoras theorem, we have,

`\therefore`|`\overline{P_2C}|^2` `=` |`\overline{P_1C}|^2``+`|`\overline{P_2C}|^2`

|`\overline{P_1P_2}|^2``=\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2`

|`\overline{P_1P_2}`| `=\sqrt{\left(x_2-x_1\right)^2+\left(y_2-y_1\right)^2}`

Note:

The distance d from origin to the point `P(x,y)` is:     `d=\sqrt{x^2+y^2}`

Collinear Points

Points lying on the same line are known as collinear points. In the following figure

A, B and C are collinear points i.e. |`\overline{AC}`|`=` |`\overline{AB}`|`+`|`\overline{BC}`|

Non-Collinear Points

Three or more points are said to be the non-collinear points if they do not lie on the same line. In the figure

A, B, and C are non-collinear points.

Note:

Three non-collinear points form a triangle, and four non-collinear points form a quadrilateral.

Mid-Point Formula

Recognize the formula to find the mid-point of the line joining two given points.

Let `A\left(x_1,y_1\right)` and `B\left(x_2,y_2\right)` be any two point of the `\overline{AB}` in the plane and `C(x, y)` be the midpoint of AB,

then `C(x, y)` `=` `\left(\frac{x_1+x_2}2,\frac{y_1+y_2}2\right)`is called mid point of A and B.