1. Coordinates
What is the Cartesian plane?
Picture a giant piece of graph paper. It has two number lines crossing each other at a right angle:
- A horizontal line called the x-axis
- A vertical line called the y-axis
They cross at a special point called the origin, where x = 0 and y = 0. This grid — the whole flat surface with these two axes on it — is called the Cartesian plane. Think of it as a map: the origin is "home base," and every other point is described by how far you've travelled from it.
y
|
6
|
(-1,4)| (2,5)
• | •
|
----------+----------- x
-6 -4 -2 0 2 4 6
|
-2
|
What do coordinates actually mean?
A coordinate is a pair of numbers, written as (x, y), that tells you exactly where a point sits on the grid:
- x tells you how far to move left/right (horizontal)
- y tells you how far to move up/down (vertical)
Memory trick
Think:
"Along the corridor, up the stairs." You always move sideways (x) first, then up or down (y). That's why it's always written (x, y) and never (y, x).
Rules for the signs:
- Positive x → move right of the origin. Negative x → move left.
- Positive y → move up from the origin. Negative y → move down.
So (2, 5) means: start at the origin, go 2 right, then 5 up. And (-1, -4) means: go 1 left, then 4 down.
Worked Example
(a) Point A is 3 squares left of the origin and 4 squares up. What are its coordinates?
- A is 3 units to the left → x = −3
- A is 4 units up → y = 4
- Write in brackets, x first: A = (−3, 4)
(b) Plot point B at (3, −2).
- Start at the origin (0, 0)
- Move 3 units right (because x = 3)
- Move 2 units down (because y = −2)
- Mark the point and label it B
Examiner tip
Always check the scale on a coordinate grid before reading off coordinates — one square doesn't always represent 1 unit. Some grids jump in 2s or 5s per square!
Practice Question 1
Point P is 5 units to the right of the origin and 3 units below it. Write down the coordinates of P.
Practice Question 2
A point Q has coordinates (−4, 0). Describe in words where Q lies, and explain what's special about it.
2. Midpoint of a Line
What is a midpoint, really?
The midpoint of a line segment is simply the point that sits exactly halfway between its two endpoints — the same distance from each end. Here's the key insight that makes this easy: the midpoint's coordinates are just the average (mean) of the two endpoints' coordinates. Average the x-values to get the midpoint's x. Average the y-values to get the midpoint's y. That's it — no geometry needed, just arithmetic.
(x1,y1) •-----------•-----------• (x2,y2)
midpoint
( x1+x2 , y1+y2 )
( 2 2 )
Worked Example
The coordinates of A are (−4, 3) and the coordinates of B are (8, −12). Find the midpoint of AB.
- Write the formula: ( (x₁+x₂)/2 , (y₁+y₂)/2 )
- Substitute: ( (−4+8)/2 , (3+(−12))/2 ) = ( 4/2 , −9/2 )
- Simplify: (2, −4.5)
Sanity check
A quick way to check your answer makes sense: the midpoint's x-value should sit between the two original x-values, and same for y. Here, 2 is between −4 and 8 ✓, and −4.5 is between 3 and −12 ✓.
Practice Question 1
Find the midpoint of the line joining C(1, 6) and D(7, 2).
Practice Question 2 (Harder — working backwards)
M(3, 1) is the midpoint of the line joining E(−1, 5) and F. Find the coordinates of F.
3. Gradient of a Line
What does "gradient" actually measure?
The gradient tells you how steep a straight line is — and which way it's tilted. Imagine walking along the line from left to right:
- If you're walking uphill, the gradient is positive.
- If you're walking downhill, the gradient is negative.
- The bigger the number (ignoring the sign), the steeper the climb or drop. A gradient of 5 is much steeper than a gradient of 2.
Concretely: a gradient of 3 means "for every 1 unit you move right, you go up 3 units." A gradient of −4 means "for every 1 unit you move right, you go down 4 units."
y y
| / |
| / uphill | \ downhill
| / (positive) | \ (negative)
| / | \
---+------- x ---+------- x
| |
How do I actually calculate it?
Pick any two points on the line and imagine (or draw) a right-angled triangle between them. Then:
If you know two exact coordinate points, (x₁, y₁) and (x₂, y₂), you can skip the diagram entirely and use the formula:
Watch out
You must subtract in a
consistent order. If you do y₂ − y₁ on the top, you must do x₂ − x₁ on the bottom (not x₁ − x₂). Mixing the order flips the sign of your answer — a classic exam slip-up.
Worked Example
(a) A line passes through (0, 2) and (1, 5). Find its gradient.
- Rise = 5 − 2 = 3 (how much y changed)
- Run = 1 − 0 = 1 (how much x changed)
- Gradient = rise/run = 3/1 = 3
- The line goes uphill, so the gradient is positive: gradient = 3
Drawing a line from a given gradient
This is the reverse skill — you're given a gradient and a point, and asked to sketch the line. Turn the gradient into a fraction (rise/run), then step it out on the grid.
Worked Example
Draw the line with gradient −2 that passes through (0, 1).
- Mark the starting point (0, 1)
- Write −2 as a fraction: −2/1 (rise = 2, run = 1)
- Negative gradient → downhill, so from (0,1) move 1 across (right) and 2 down
- Mark that new point, then draw a straight line through both points
Handy shortcut
If the gradient is a whole number like −5, just write it as a fraction over 1: −5/1. Then rise = 5, run = 1.
Practice Question 1
Find the gradient of the line joining points (−2, −3) and (4, 9).
Practice Question 2
A line has gradient 3/4 and passes through (−1, −2). Describe how you'd draw it (rise, run, and direction of each step).
4. Length of a Line
Where does this formula come from?
To find the straight-line distance between two points, we secretly use Pythagoras' theorem (a² + b² = c²). Here's the idea: if you draw a right-angled triangle where the line segment is the hypotenuse, then:
- The horizontal leg of the triangle is the difference in x-coordinates
- The vertical leg is the difference in y-coordinates
- The line segment itself is the hypotenuse — the distance you want
(x1,y1)
•-------------• b (horizontal leg)
| /|
| / |
a | / | c (vertical leg)
| /
| /
| •
(x2,y2)
distance² = b² + c²
Why the order doesn't matter here
Unlike gradient, it's fine to do either (x₁ − x₂) or (x₂ − x₁) — because you square the result straight after, and squaring always makes negatives positive. So the order of subtraction can't mess up your final answer.
Worked Example
Point A has coordinates (3, −4) and point B has coordinates (−5, 2). Calculate the length of AB.
- Formula: d = √[(x₁−x₂)² + (y₁−y₂)²]
- Substitute: d = √[(3 − (−5))² + ((−4) − 2)²]
- Be careful with negatives: 3 − (−5) = 8, and (−4) − 2 = −6
- d = √[8² + (−6)²] = √[64 + 36] = √100 = 10 units
Examiner tip
Put negative numbers in brackets before subtracting to avoid sign errors, e.g. write it as
(−6) − (−8) = 2 rather than trying to do it in your head. This is one of the most common places marks are dropped.
Practice Question 1
Find the distance between the points (1, 2) and (4, 6).
Practice Question 2
The distance between points (−2, k) and (2, 1) is 5 units. Find the possible value(s) of k.