## Sunday, August 1, 2010

### Questions

Go to the following URLs to test your understanding of coordinate geometry:

http://www.tutorvista.com/q/using-geometry-coordinate-geometry/86682/

http://www.teachnet-uk.org.uk/2007%20projects/maths-c2_worksheets/c2%20pure%20maths%20worksheets/C2-Chapter4.htm

### Formulas

The Gradient of a Line Joining Two Points

Apply the following formula when finding the gradient of the line joining the points (x1, y1) and (x2, y2):

y2 - y1/ x2 - x1

Example:

Find the gradient of the line joining the points (3, 2) and (4, 6).

Gradient = 6 - 2/4 - 3= 4/1 = 4

The Midpoint of a Line Joining Two Points

When finding the midpoint of the line joining the points (x1, y1) and (x2, y2) apply the following formula:

[½(x1 + x2), ½(y1 + y2)]

Example

Find the coordinates of the midpoint of the line joining (3, 2) and (5, 3).

Midpoint = [½(5 + 3), ½(2 + 3)] = (4, 2.5)


Parallel and Perpendicular Lines

When two lines are parallel, they have the same gradient.
When two lines are perpendicular, the product of the gradients of the two lines are -1.

Example:

a) y = 2x + 1
b) y = -½ x + 2
c) ½y = x - 3

The gradients of the lines are 2, -½ and 2 respectively. Therefore (a) and (b) and perpendicular, (b) and (c) are perpendicular and (a) and (c) are parallel.

The Equation of a Line Using One Point and the Gradient

The equation of a line which has gradient m and which passes through the point (x1, y1) is:

y - y1 = m(x - x1)

The equation of a line passing through (x1, y1) and (x2, y2) can be written as:

y - y1 = y2 - y1

x - x1 x2 - x1

References:
http://www.onlinemathlearning.com/coordinate-geometry.html
http://www.mathsrevision.net/alevel/pages.php?page=6

### Real life application of coordinate geometry

Coordinate geometry is applied to many professions and used by many in real life. For example it can be used by computer programmers. Computer programmers use coordinate geometry because most of the programs that they write generate PDF files. And in a PDF file, the printed page is one big coordinate plane. Coordinate geometry is thus used to position elements on the page. PDF files that are produced contains text, images and line drawings, all of which are placed into position by using (x,y) coordinates, distances, slopes, and simple trigonometry.

Coordinate geometry are also used in manipulating images. The selected image is like a big coordinate plane with each colour information as each individual points. Thus when the colours of the pictures are being manipulated, the points are changed.

Coordinate geometry are also applied in scanners. Scanners make use of coordinate geometry to reproduce the exact image of the selected picture in the computer. It manipulates the points of each information in the original documents and reproduces them in soft copy.

Thus coordinate geometry are widely used without our knowing. They are mostly commonly used in reproducing the original and also in IT forms.

References:
http://images.amazon.com/images/G/01/software/detail-page/B000LU89Y8-4.jpg
http://wpcontent.answcdn.com/wikipedia/commons/6/6e/Rgb-raster-image.png

### Introduction

Definition

Coordinate geometry refers to:

A system of geometry where the position of points on the plane is described using a pair of numbers.

A plane is a flat surface that goes on forever in both directions thus coordinate geometry is able to tell the exact location where a point is.

The above diagram shows the parts of the coordinate graph. A point on the graph is defined by two given numbers, one is the point on the x-axis and the other is the point on the y-axis. Together, they represent a unique position on the plane.

Thus take for example point K on the graph. Point K has an x value of 3 and a y value of 4. These are the coordinates or sometimes referred to as the rectangular coordinates of point K.

History

The method of describing the location of points in this way was proposed by the French mathematician René Descartes (1596 - 1650). He also made one of the greatest advances in geometry by fusing algebra and geometry. He had even proposed further that curves and lines could be described by equations using this technique. His work’s are thus known as Cartesian coordinates, and its coordinate plane as the Cartesian Coordinate Plane. A myth has it that he was watching a fly on the ceiling when he conceived of locating points on a plane with a pair of numbers. Another mathematician that has also discovered coordinate geometry is Fermat, however the geometry that we are using today is the Descartes version.

References:
http://math.rice.edu/~lanius/Geom/his.html

http://www.ricksmath.com/images/coordinates-point.jpg