Coordinate Geometry
Coordinate Geometry
Coordinate Geometry tests points in the Cartesian plane, distance, midpoint, gradient, straight-line equations, parallel and perpendicular lines, and simple intersections.
Core rules
- Distance between (x₁, y₁) and (x₂, y₂) is √[(x₂ − x₁)² + (y₂ − y₁)²].
- Midpoint is ((x₁ + x₂)/2, (y₁ + y₂)/2).
- Gradient m = (y₂ − y₁)/(x₂ − x₁). A line y = mx + c has gradient m and y-intercept c.
- Parallel lines have equal gradients. Perpendicular gradients multiply to −1, so the perpendicular gradient is the negative reciprocal.
- To find where two lines meet, solve their equations simultaneously.
Worked example
For A(2, 3) and B(8, 11): midpoint = ((2+8)/2, (3+11)/2) = (5, 7). Gradient = (11−3)/(8−2) = 8/6 = 4/3. Distance = √[(8−2)² + (11−3)²] = √(36 + 64) = √100 = 10. A line perpendicular to AB has gradient −3/4. The line through A with that gradient is y − 3 = −3/4(x − 2).
Traps
- Subtracting coordinates in the wrong order and losing the sign of the gradient.
- Using the reciprocal rather than the negative reciprocal for a perpendicular gradient.
- Forgetting that horizontal lines have gradient 0 and vertical lines have undefined gradient.
- Substituting a point into y = mx + c but mixing up x and y.