Co-ordinate Geometry of the Line

Co-ordinate Geometry is a relatively modern and immensely useful branch of mathematics. The idea of giving points in the plane co-ordinates makes it much easier to deal with many properties of geometry that had previously been tackled using so-called Euclidean geometry (i.e., theorems).

Coordinates are pairs of numbers that are used to determine points in a plane, relative to a special point called the origin. The origin has coordinates (0, 0).

One fundamental idea in co-ordinate geometry is that of the equation of a line. In this topic, we examine the idea of the equation of a line and its properties, e.g., slope. We also consider other basic concepts such as distance, midpoint and the area of a triangle.

It is also important to note that many of the ideas from this topic come into many others, e.g., the circle, graphs and linear programming. There is also a very close link between the Argand diagram in Complex Numbers and co-ordinate geometry.

For this course the  study of the Line can be divided into the following sections:

Distance and Midpoint

Only used to get the distance between two known points

If you draw this out on a graph so that a right angle triangle is formed by the line segment and lines parallel to the X and Y axis through the points at each end of the line segment you can see that this is actually an application of Pythagoras’ theorem.

Note that the midpoint should be written as a point

Slope of a Line
Learn the formula exactly
We get one y on the LHS
Change the sign and turn upside down
You must get the slope of each line first

If lines are parallel, their slopes are the same.

Equation of a Line
Learn the formula exactly
Either the statement is true or it is not
Let y=0 and let x=0
Same method as in algebra
Get the slope and a point on the line

Area of a Triangle
A diagram is often useful here
Learn the formula exactly
It does not matter which point is translated to the origin
This site includes some material relevant to coordinate geometry. Each link brings you through to a number of questions on that topic, and by clicking on the question number, you are shown a worked solution of that question.