Coordinates in all four quadrants
Coordinates underpin every graph question on WJEC. Foundation candidates plot and read points; Intermediate and Higher candidates use distance and midpoint formulae.
The Cartesian plane
The horizontal axis is x, the vertical is y. The origin O is (0, 0). Coordinates are written (x, y).
The four quadrants:
- Quadrant I (top-right): x > 0, y > 0
- Quadrant II (top-left): x < 0, y > 0
- Quadrant III (bottom-left): x < 0, y < 0
- Quadrant IV (bottom-right): x > 0, y < 0
Points on the x-axis have y = 0 (e.g. (5, 0)). Points on the y-axis have x = 0 (e.g. (0, −3)).
Midpoint of a line segment
For points A(x₁, y₁) and B(x₂, y₂):
midpoint M = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)
Example: midpoint of (−2, 5) and (4, 1) is ((−2 + 4)/2, (5 + 1)/2) = (1, 3).
Distance between two points
distance = √((x₂ − x₁)² + (y₂ − y₁)²)
This is Pythagoras' theorem applied to the horizontal and vertical changes. WJEC Higher Unit 1 expects exact (surd) answers.
Example: distance from (1, 2) to (4, 6) is √((4 − 1)² + (6 − 2)²) = √(9 + 16) = √25 = 5.
Translation by a vector
Adding (a, b) to a coordinate translates by a in x and b in y. Useful for setting up shape transformations.
WJEC exam tip
When using the midpoint or distance formulae with negative coordinates, write subtractions in brackets BEFORE evaluating. (4 − (−2))² = 6² is the safe path; rushing to "4 − −2" without brackets is the most common Foundation slip.
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