Coordinate Geometry (AQA GCSE Maths)
Revision Note
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Maths
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;
-
- Midpoint of a Line
- Gradient of a Line
- Length of a Line
Midpoint 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.
- E.g. dividing AB in the ratio 2 : 3 means finding the point that is
- 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
Exam Tip
- If working in 2D (most questions!) making a quick sketch of the two points will help you know roughly where the midpoint should be, which can be helpful to check your answer
- If working in 3D (some questions!), just check that your midpoint's
coordinate lies between the two given x coordinates, and so on for the
and
coordinates
Worked example
The coordinates of A are (−4, 3) and the coordinates of B are (8, −12).
(a)
Find M, the midpoint of AB.
The midpoint can be found using M =
Fill in the values of x and y from each coordinate
Simplify
M = (2, −4.5)
(b)
A point N divides AB in the ratio 1 : 2.
Find the coordinates of N.
Calculate the difference between the x coordinate of A and the x coordinate of B
Divide this difference by 3 (as there are 1 + 2 = 3 parts in the ratio 1 : 2)
Add 4 to the x coordinate of A
Repeat for y
Alternatively, you may find it helpful to sketch the coordinates A and B and find N intuitively. Note that in the sketch, the coordinates do not need to be placed in the correct orientation to one another, simply along a straight line:
![2-12-2-midpoints-we](https://cdn.savemyexams.com/cdn-cgi/image/f=auto,width=1920/uploads/2022/11/2-12-2-midpoints-we.png)
Write the final answer as a coordinate point
N = (0, −2)
Gradient of a Line
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
- As
- 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.)
- For every 1 unit to the right, the line moves upwards by
- 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
Exam Tip
- Be very careful with negative numbers when calculating the gradient; write down your working rather than trying to do it in your head to avoid mistakes
- For example,
- For example,
Worked example
(a)
Find the gradient of the line joining (-1, 4) and (7, 28)
Using :
Simplify:
Gradient = 3
(b)
On the grid below, draw a line with gradient −2 that passes through (0, 1).
Mark the point (0, 1) and then count 2 units down for every 1 unit across
(c)
On the grid below, draw a line with gradient that passes through (0,-1)
Mark the point (0,-1) and then count 2 units up for every 3 units across
Length of a 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
Exam Tip
- As we are squaring the difference in and in the formula, it does not matter if they are positive or negative
- 32 is the same as (-3)2, this may help to speed up your working
Worked example
Point A has coordinates (3, -4) and point B has coordinates (-5, 2).
Calculate the distance of the line segment AB.
Using the formula for the distance between two points,
Substituting in the two given coordinates:
Simplify:
Answer = 10 units
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