Coordinates
The Cartesian Plane
The Cartesian plane is a two-dimensional grid system used to locate points. It consists of:
- Horizontal axis (x-axis): Runs left and right
- Vertical axis (y-axis): Runs up and down
- Origin: The point where both axes meet, at (0, 0)
Coordinates (x, y) are a pair of numbers that pinpoint exactly where a point is located on the grid. The first number is the x-coordinate (horizontal position), and the second is the y-coordinate (vertical position).
Understanding Direction
The mnemonic "Along the corridor, up the stairs" helps you remember: x first (horizontal), then y (vertical).
✓ Worked Example: Reading Coordinates
Find the coordinates of point A at 3 units left and 4 units up from the origin.
• 3 units left means x = −3 (negative, because it's to the left)
• 4 units up means y = 4 (positive, because it's above the origin)
Answer: A = (−3, 4)
Try This
What are the coordinates of a point that is 2 units to the right and 5 units down from the origin?
📌 Examiner's Tip: Always check the scale on the coordinate grid. One square doesn't always equal 1 unit—the grid might be scaled differently.
Four Quadrants
The plane is divided into four regions:
- Quadrant I (top right): x positive, y positive
- Quadrant II (top left): x negative, y positive
- Quadrant III (bottom left): x negative, y negative
- Quadrant IV (bottom right): x positive, y negative
Midpoint of a Line
What Is a Midpoint?
The midpoint is the point exactly halfway between two endpoints on a line. It's equidistant from both points. Think of it as the "average location" of two points—you're finding the average of the x-coordinates and the average of the y-coordinates separately.
Midpoint = ( (x₁ + x₂)/2 , (y₁ + y₂)/2 )
How to Use the Formula
- Add the two x-coordinates together and divide by 2
- Add the two y-coordinates together and divide by 2
- Write your answer as a coordinate pair
✓ Worked Example: Finding a Midpoint
Find the midpoint of the line segment AB where A = (−4, 3) and B = (8, −12).
Step 1: Average the x-coordinates
x = (−4 + 8) / 2 = 4 / 2 = 2
Step 2: Average the y-coordinates
y = (3 + (−12)) / 2 = −9 / 2 = −4.5
Answer: Midpoint = (2, −4.5)
Try This
Find the midpoint of the line joining P = (6, 2) and Q = (−2, 8).
⚠️ Common Mistake: Students often forget to divide by 2 after adding the coordinates. Remember: midpoint means the AVERAGE, so you must divide by 2.
Gradient of a Line
What Is Gradient?
The gradient (or slope) measures how steep a line is. It describes how much the line rises (or falls) for every unit it moves to the right.
Positive gradient: The line slopes upward from left to right (uphill). A gradient of 3 means for every 1 unit right, you go up 3 units.
Negative gradient:
Gradient Formula
Length of a Line