Coordinates in all four quadrants
Coordinates are the bridge between geometry and algebra. The Cartesian plane has two perpendicular axes (the x-axis and y-axis) crossing at the origin (0, 0). Every point is described by an ordered pair (x, y) — x first (horizontal), y second (vertical).
The four quadrants
Numbered anti-clockwise starting top-right:
- Q1: x > 0, y > 0 (top-right) — e.g. (3, 4)
- Q2: x < 0, y > 0 (top-left) — e.g. (-3, 4)
- Q3: x < 0, y < 0 (bottom-left) — e.g. (-3, -4)
- Q4: x > 0, y < 0 (bottom-right) — e.g. (3, -4)
Points on the axes themselves belong to no quadrant.
Plotting and reading
Always count along the x-axis first, then up/down the y-axis. Make sure your plot lands at a crossing point of grid lines, not in the middle of a square.
Distance between two points
Given A(x₁, y₁) and B(x₂, y₂), the distance is from Pythagoras' theorem:
AB = √((x₂ - x₁)² + (y₂ - y₁)²)
Example: distance between A(1, 2) and B(4, 6).
Differences: 4 - 1 = 3, 6 - 2 = 4. Distance = √(9 + 16) = √25 = 5.
Midpoint of a segment
Average the coordinates:
M = ((x₁ + x₂)/2, (y₁ + y₂)/2).
Example: midpoint of (1, 2) and (5, 8) is ((1 + 5)/2, (2 + 8)/2) = (3, 5).
Gradient between two points
gradient = (y₂ - y₁)/(x₂ - x₁) — the rise over run.
Example: from (1, 2) to (4, 8): gradient = (8 - 2)/(4 - 1) = 6/3 = 2. The line goes up 2 units for every 1 unit right.
✦Worked example— Worked example — combining
A is (-2, 3); B is (4, -5). Find (i) midpoint, (ii) length AB, (iii) gradient AB.
- Midpoint: ((-2 + 4)/2, (3 + -5)/2) = (1, -1).
- Length: √((4 - (-2))² + (-5 - 3)²) = √(36 + 64) = √100 = 10.
- Gradient: (-5 - 3)/(4 - (-2)) = -8/6 = -4/3.
⚠Common mistakes— Common mistakes (examiner traps)
- Reversing x and y. "Along the corridor and up the stairs" — x first, y second.
- Sign errors with negatives.
(4 - (-2)) = 6, not 2. Use brackets when subtracting negatives. - Forgetting the square root in the distance formula. The formula gives distance squared until you root it.
- Using midpoint = (x₂ - x₁)/2. That's half the run, not a midpoint. Use the SUM, divided by 2.
- Confusing rise/run. The gradient is vertical change ÷ horizontal change, in that order.
➜Try this— Quick check
(a) Midpoint of (-3, 7) and (5, 1): ((−3 + 5)/2, (7 + 1)/2) = (1, 4). (b) Distance between (0, 0) and (5, 12): √(25 + 144) = √169 = 13.
AI-generated · claude-opus-4-7 · v3-deep-algebra