Math Guide

Maths Project on Coordinate Geometry

Introduction:

                                  Rene Descartes, a French Mathematician introduced "coordinate geometry".It is often referred as "analytical geometry". Descartes initially made two famous discoveries.

They are

1) any ordered pair of numbers can be represented by a point in a plane.

2) any curve in the two-dimensional space can be identified by an equation in x and y.

Thecoordinate plane forms the basis of coordinate geometry. The co-ordinate plane is a two-dimensional plane in terms of two axes: x andy. The x-axis indicates the horizontal axis while the y-axis indicates the vertical axis of the plane. The x-axis and y-axis are perpendicular to each other and meet at a point called "origin" .In the coordinate plane, points are indicated by ordered pairs in which first coordinate is x- coordinate also called "abscissa" and second coordinate is y-coordinate also called "ordinate".


Concepts of coordinate geometry:


Slope:

                      On the coordinate plane, the slant of a line is called the slope. Slope is the ratio of the change in the y-value over the change in the x-value. Given any two points on a line, you can calculate the slope ofthe line by using this formula:

                                               slope = changeiny/changeinx

For ex:

  Given two points, P = (0, –1) and Q = (4,1), on the line we can calculate the slope of the line.

                                                slopeslopevalues

  •    Slope of x-axis is 0 and slope of any line parallel to x-axis is zero .
  •    Slope of y-axis is not defined and slope of any line parallel to y-axis is also undefined.


General Form of Line Equations:

  •   Slope-Intercept Form:

                                   In coordinate geometry, the equation of a line can be written in the form, y = mx + b, where m is the slope and b is the y-intercept. 

                                                                       y = mx+b

Ex: For the equation, y=1/2x-1 slope is 1/2 and y-intercept is -1. 

  • Point Slope Form:

                                  The equation of the line passing through a given point (x1,y1) and having slope 'm' is:

                                                                     y - y1 = m (x - x1).

Slope of Parallel Lines:

                      Two lines are said to be parallel if they have same slope.

 Example: The line lineequation is parallel to the line lineequation because slopes (1/2) are same.

Slope of Perpendicular Lines:

                                Two lines are said to be perpendicular if the product of their slopes is -1.

Example: The line lineequation is perpendicular to the line y = –2x – 1because the product of the two slopes is 1/2*-2.


Mid-point formula and distance formula of coordinate geometry:


Mid-Point Formula:

                 Mid point that is halfway between two given points, get the average ofthe x-values and the average of the y-values.

The midpoint between the two points (x1,y1) and (x2,y2) is midpoint 

Ex: The midpoint of the points A(1,4) and B(5,6) is
    midpoint.

Distance Formula:

                     The distance between the two points (x1,y1) and (x2,y2) is
  

Ex: Distance between A(1,1) and B(3,4) is 

                                                    linesegmentnotation2 = 22 + 32
                                                    linesegmentnotation2 = 13

                                                                    linesegmentnotation = sqroot13.