Quick Summary
- Coordinates — an (x, y) pair pins down the exact location of a point on the Cartesian plane.
- Midpoint — the exact middle of a line segment; found by averaging the x's and averaging the y's.
- Gradient — how steep a line is; found using "rise over run" or the gradient formula.
- Length of a line — the straight-line distance between two points; found using Pythagoras' theorem in disguise.
- All four ideas connect: once you can read coordinates, midpoint/gradient/length are just formulas applied to two (x, y) pairs.
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?
midpoint
average (mean)
3. Gradient of a Line
What does "gradient" actually measure?
gradient
- uphill
positive
- downhill
- steeper