Coordinate Geometry
Coordinate geometry in SPM Additional Mathematics moves between algebra and the Cartesian plane: distance, midpoint, gradient, equations of straight lines, parallel and perpendicular lines, areas of polygons and loci. The exam rewards clean formula use and punishes sign slips.
Formulas
Distance: √[(x₂−x₁)² + (y₂−y₁)²]. Midpoint: ((x₁+x₂)/2, (y₁+y₂)/2). Gradient: m = (y₂−y₁)/(x₂−x₁). Parallel lines have m₁ = m₂; perpendicular lines satisfy m₁m₂ = −1. The line through (x₁, y₁) with gradient m is y − y₁ = m(x − x₁). The area of a polygon with vertices taken in order is ½|Σxᵢyᵢ₊₁ − Σyᵢxᵢ₊₁|.
Worked example
Find the equation of the line through (2, −1) perpendicular to 3x + 4y = 8. Rewrite as y = −(3/4)x + 2, so the gradient is −3/4 and the perpendicular gradient is 4/3. Then y + 1 = (4/3)(x − 2), so 3y + 3 = 4x − 8 and 4x − 3y − 11 = 0. Check: at (2, −1), 8 + 3 − 11 = 0.
Traps
Using the same gradient rather than the negative reciprocal; omitting the absolute value in the area formula; listing vertices out of order; confusing midpoint with distance.