Coordinate geometry problems

WJEC blends algebra and geometry on coordinate axes. The toolkit is small but you must apply it confidently.

Distance between two points

Given A(x₁, y₁) and B(x₂, y₂): distance AB = √((x₂ − x₁)² + (y₂ − y₁)²).

This is Pythagoras applied to the horizontal and vertical separations.

Midpoint of a segment

midpoint M = ((x₁ + x₂)/2, (y₁ + y₂)/2). Average the x's, average the y's.

Gradient of a line through two points

m = (y₂ − y₁) / (x₂ − x₁).

Equation of a straight line

y − y₁ = m(x − x₁) — point-gradient form. Or y = mx + c with c found by substitution.

Parallel and perpendicular lines

• Parallel lines have equal gradients.
• Perpendicular gradients multiply to give −1: m₁ × m₂ = −1, so m₂ = −1/m₁.

Common WJEC question patterns

1. Show three points are collinear: compute two gradients between pairs; if equal, they lie on the same line.
2. Show a triangle is right-angled: check whether two sides have perpendicular gradients (product = −1).
3. Find where a line meets an axis: set y = 0 (x-intercept) or x = 0 (y-intercept).

Worked check — perpendicular bisector

Perpendicular bisector of AB:

1. Find midpoint M.
2. Find gradient of AB; take negative reciprocal for perpendicular gradient.
3. Use point-gradient form through M.

WJEC exam tip

Always sketch a quick diagram, even when not asked. Marking schemes credit a correct method based on a labelled sketch M1 before any arithmetic, which protects you when the calculator slips.