Geometric problems on coordinate axes
This topic combines coordinate geometry with shape properties — finding distances, midpoints, gradients and equations of lines through pairs of points.
Distance between two points
Distance = √((x₂ − x₁)² + (y₂ − y₁)²) — Pythagoras applied to coordinates.
Worked example: distance from (1, 2) to (5, 6).
- = √(4² + 4²) = √32 = 4√2 ≈ 5.66.
Midpoint of two points
Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2) — average of coordinates.
Worked example: midpoint of (3, 1) and (7, 9). = (5, 5).
Gradient
Gradient m = (y₂ − y₁) / (x₂ − x₁).
Vertical lines have undefined gradient; horizontal lines have gradient 0.
Equation of a line through two points
- Find gradient m.
- Use y − y₁ = m(x − x₁) with one point.
- Rearrange to y = mx + c if needed.
Worked example: line through (1, 3) and (4, 9).
- m = (9 − 3)/(4 − 1) = 2.
- y − 3 = 2(x − 1) → y = 2x + 1.
Parallel and perpendicular gradients
- Parallel lines have the SAME gradient.
- Perpendicular gradients multiply to −1: m₁ × m₂ = −1, or m₂ = −1/m₁.
Worked example: line perpendicular to y = 3x + 2, passing through (4, 5).
- New gradient: −1/3.
- y − 5 = −1/3 (x − 4) → y = −x/3 + 4/3 + 5 = −x/3 + 19/3.
Special quadrilaterals on axes
To prove a quadrilateral is a parallelogram: show two pairs of opposite sides have equal gradients (parallel) and equal lengths. To prove a rhombus: parallelogram + all sides equal. To prove a square: rhombus + sides perpendicular.
⚠Common mistakes
- Distance formula sign errors — always square the differences.
- Midpoint formula confused with gradient.
- Perpendicular gradient flip-only — must also negate.
- Forgetting integer simplification — keep exact values when possible.
- Reading the wrong axis — x is horizontal, y vertical.
➜Try this— Quick check
Find midpoint and length of segment from A(−2, 1) to B(4, 9).
- Midpoint = (1, 5).
- Length = √(36 + 64) = √100 = 10.
AI-generated · claude-opus-4-7 · v3-deep-geometry