Coordinate Geometry | Edexcel GCSE Maths: Foundation Revision Notes 2017

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 $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ is
- $(\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2})$

Exam Tip

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.

Worked example

The coordinates of A are (−4, 3) and the coordinates of B are (8, −12).

Find M, the midpoint of AB.

Sketch a diagram

Straight line between points A and B
The midpoint can be found using M = $(\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2})$

Fill in the values of x and y from each coordinate

$(\frac{- 4 + 8}{2}, \frac{3 + - 12}{2}) = (\frac{4}{2}, \frac{- 9}{2})$

Simplify

M = (2, −4.5)

Gradient of a Line

What is the gradient of a line?

The gradient is a measure of how steep a straight line is
A gradient of 3 means:
- For every 1 unit to the right, go up by 3
A gradient of -4 means:
- For every 1 unit to the right, go down by 4
A gradient of 3 is steeper than 2
- A gradient of -5 is steeper than -4
A positive gradient means the line goes upwards (uphill)
- Bottom left to top right
A negative gradient means the line goes downwards (downhill)
- Top left to bottom right

How do I find the gradient of a line?

Find two points on the line and draw a right-angled triangle
- Then $gradient = \frac{change in y}{change in x}$
- Or, in short,
  - The rise is the vertical length of the triangle
  - The run is the horizontal length of the triangle
- Put the correct sign on your answer
  - Positive for uphill lines
  - Negative for downhill lines
- You can also find gradient of a line between two points, and
  - Use the formula $\frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

How do I draw a line with a given gradient?

To draw the gradient
- The rise is 2
- The run is 3
- It is positive (uphill)
  - Move 3 units to the right and 2 units up
To draw the gradient make it a fraction,
- The rise is 5
- The run is 1
- It is negative (downhill)
  - Move 1 unit to the right and 5 units down

Exam Tip

Remember to make your gradient negative for downhill lines!
Make sure that you draw a straight line to the edges of the graph.
- Don't just join up two points to create a line segment.

Worked example

(a)

Find the gradient of the line shown in the diagram below.

screenshot-2023-02-12-at-20-42-17
Find two points that the line passes through

$(0, 2) and (1, 5)$

Use the grid to draw a right-angled triangle
Find the 'rise' (vertical length) and 'run' (horizontal length)

cie-igcse-core-gradient-of-a-line-rn-we-a

Work out the fraction $\frac{rise}{run}$

$\frac{3}{1} = 3$

Look to see if the line is uphill or downhill

uphill, so the gradient is positive

The gradient is 3

(b)

On the grid below, draw the line with a gradient of −2 that passes through (0, 1).

Mark on the point (0, 1)
-2 is the fraction $- \frac{2}{1}$
The rise is 2, the run is 1, the line goes downhill (so 1 across, 2 down)

Mark on the next point (1, -1) and draw a straight line between them

cie-igcse-gradients-of-lines-we-1

On the grid below, draw the line with a gradient of $\frac{2}{3}$ that passes through (0, -1).

Mark on the point (0, -1)

The rise is 2, the run is 3, the line goes uphill (so 3 across, 2 up)

Mark on the next point (3, 1) and draw a straight line through the two points

cie-igcse-gradients-of-lines-we-2

Coordinate Geometry (Edexcel GCSE Maths: Foundation)

Revision Note

Midpoint of a Line

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

Exam Tip

Worked example

Gradient of a Line

What is the gradient of a line?

How do I find the gradient of a line?

How do I draw a line with a given gradient?

Exam Tip

Worked example

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Author: Naomi C