Coordinates in 2D
Coordinates in 2D
Coordinate geometry is a method of analysing geometric shapes.
To start off with, let us consider one of the simplest geometric problems: describing the position of points on a 2-dimensional surface (such as a piece of paper).
The position of points will usually be referred to in one of these three ways:
- Fixed point: for example, (2,6).
- Generic fixed points: for example, (x1, y1), (x2, y2).
- General points: for example, (x, y). This is telling us that the point can be anywhere along a shape.
Using coordinates means that we have to define where we start measuring from. When we are defining points on a graph, our point of reference is nearly always the origin or (0, 0).
The next thing to be happy with is how to calculate the distance between any 2 points.
We are interested in finding the distance from A to B. A is at (x1, y1) and B is at (x2, y2). Let's take a look at this using the diagram below:
We want to find the distance between A and B. If you look at the diagram above, we have added an imaginary point L, so that the triangle ABL is a right-angled triangle.
Using Pythagoras' Theorem we can now express the length of the line AB in terms of A, B and L:
AB2 = AL2 + BL2
Remember that the position of A is (x1, y1) and the position of B is (x2, y2). So the distance between A and L is x2 - x1 and the distance between B and L is y2 - y1 (see diagram above).
Hence, we can re-write the above equation as:
AB2 = (x2 - x1)2 + (y2 - y1)2
Solving this for AB gives us the equation we are trying to prove, telling us that the distance between A and B can be expressed by:
So now you can find the distance between any two points (x1, y1) and (x2, y2), and also understand how the formula was derived!
Here's a quick example of how this would work with numbers.
The distance between (2,6) and (-3,9) is:
The next step is to learn how to calculate the midpoint, M, between any two points.
Consider this diagram:
M is the midpoint of A and B.
The x coordinate of the midpoint M is found by taking the average of the x coordinate of the points A and B (this is half way between x1 and x2 on the diagram above).
The y coordinate of the midpoint M is also found by taking the average of the y coordinates of the points A and B (this is half way between y1 and y2 on the diagram).
Therefore, we can say that:
The x coordinate of M equals
The y coordinate of M equals
So using this technique you can find the midpoint of any line defined by two points. e.g. the midpoint of (2,6) and (-3,9) is (-0.5, 7.5)
The gradient of a straight line measures its slope in relation to the x-axis. In other words, the amount that y increases for every unit increase in x.
The gradient of a line can be positive or negative.
A positive gradient has is an 'uphill' slope in relation to the positive x-axis, and this means that a positive increase in the x direction is accompanied by a positive increase in the y direction.
A negative gradient has a 'downhill' slope in relation to the positive x-axis. This means that a positive increase in the x direction is accompanied by a decrease in the y direction.
How to calculate the gradient of a line
The gradient of a line is the increase in the y coordinate divided by the increase in the x coordinate between any 2 points on a line.
Looking at the graph we can say that:
Note: This method can only be used to find the gradient of a straight line. If you need to find the gradient of a point on a curve, you need to use differentation (see the Differentiation Learn-It. There is a whole S-cool! topic dedicated to differentiation.)