Coordinate Geometry (CIE IGCSE Maths: Extended)

What is coordinate geometry?

• Coordinate geometry is the study of geometric figures like lines and shapes, using coordinates.
• Given two points, at GCSE, you are expected to know how to find;

1. 1. Midpoint of a Line
   2. Gradient of a Line
   3. Length of a Line

How do I find the midpoint of a line in two dimensions (2D)?

• The midpoint of a line will be the same distance from both endpoints
• You can think of a midpoint as being the average (mean) of two coordinates
• The midpoint of  and is

How can I extend the idea of the midpoint in two dimensions (2D)?

• The midpoint of AB splits AB in the ratio 1 : 1.
• If you are asked to find the point that divides AB in the ratio m : n, then you need to find the point that lies of the way from A to B.
  • E.g. dividing AB in the ratio 2 : 3 means finding the point that is of the way from A to B.
• Normally an exam question will ask you to find the point that divides AB in the ratio 1 : n, so;
  • find the difference between the x coordinates,
  • divide this difference by [1 + n], and add the result to the x coordinate of A,
  • repeat for the y coordinates.

How do I find the midpoint of a line in three dimensions (3D)?

• Finding the midpoint of a line in 3D involves a simple extension of the formula used for 2D midpoints
• Similar to before, you can think of a midpoint as being the average (mean) of three and three coordinates
• The midpoint of  and is

What is the gradient of a line?

• The gradient is a measure of how steep a 2D line is
• A large value for the gradient means the line is steeper than for a small value of the gradient
• A gradient of 3 is steeper than a gradient of 2
• A gradient of −5 is steeper than a gradient of −4
• A positive gradient means the line goes upwards from left to right
• A negative gradient means the line goes downwards from left to right
• In the equation for a straight line, , the gradient is represented by 
• The gradient of is −3

How do I find the gradient of a line?

• The gradient can be calculated using

• You may see this written as  instead
• For two coordinates  and  the gradient of the line joining them is

• The order of the coordinates must be consistent on the top and bottom
• i.e. (Point 1 – Point 2) or (Point 2 – Point 1) for both the top and bottom

How do I draw a line with a given gradient?

• A line with a gradient of 4 could instead be written as . 
  • As  , this would mean for every 1 unit to the right ( direction), the line moves upwards ( direction) by 4 units.
  • Notice that 4 also equals , so for every 1 unit to the left, the line moves downwards by 4 units
• If the gradient was −4, then or . This means the line would move downwards by 4 units for every 1 unit to the right.
• If the gradient is a fraction, for example , we can think of this as either
  • For every 1 unit to the right, the line moves upwards by , or
  • For every 3 units to the right, the line moves upwards by 2.
  • (Or for every 3 units to the left, the line moves downwards by 2.)
• If the gradient was  this would mean the line would move downwards by 2 units for every 3 units to the right
• Once you know this, you can select a point (usually given, for example the -intercept) and then count across and upwards or downwards to find another point on the line, and then join them with a straight line

How do I calculate the length of a line?

• The distance between two points with coordinates  and  can be found using the formula

• This formula is really just Pythagoras’ Theorem  , applied to the difference in the -coordinates and the difference in the -coordinates;

• You may be asked to find the length of a diagonal in 3D space. This can be answered using 3D Pythagoras