Geometric problems on coordinate axes

Edexcel 1MA1 examines coordinate geometry across all three papers. Foundation tier focuses on midpoints and distances; Higher tier extends to perpendicular bisectors, equation-of-line problems, and proofs that a quadrilateral is a particular shape.

Core formulae

For points A(x₁, y₁) and B(x₂, y₂):

• Midpoint M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
• Length AB = √((x₂ − x₁)² + (y₂ − y₁)²) — Pythagoras applied to the coordinate triangle.
• Gradient m = (y₂ − y₁) / (x₂ − x₁)

Parallel and perpendicular

• Parallel lines have equal gradients.
• Perpendicular lines have gradients whose product is −1 (negative reciprocals: m₁ × m₂ = −1).

Equation of a straight line

y = mx + c, where m is the gradient and c is the y-intercept. Given a gradient m and a point (x₀, y₀), use y − y₀ = m(x − x₀).

Identifying shapes

To prove a quadrilateral is, e.g., a parallelogram:

• Show opposite sides have equal gradients (parallel) and equal lengths (the second part is what makes it more than a trapezium).

For a rhombus: all four sides equal length. For a rectangle: opposite sides parallel AND adjacent sides perpendicular.

Common Edexcel mark-scheme phrasing

• M1 for a correct gradient or length expression.
• A1 for a correct simplified value.
• B1 for a correctly identified property (e.g. "AB parallel to DC because gradients equal").
• QWC on "show that" or "prove" items — pupils must state what is being compared and conclude.