Coordinate Geometry and Straight Lines
Coordinate Geometry and Straight Lines tests the distance formula, section formula, slope, and the standard forms of a line. A diagram is not supplied, so sign discipline is essential.
Core rules
- Distance between (x₁, y₁) and (x₂, y₂): √[(x₂ − x₁)² + (y₂ − y₁)²].
- Midpoint: ((x₁ + x₂)/2, (y₁ + y₂)/2). Section formula for ratio m:n: ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)).
- Slope m = (y₂ − y₁)/(x₂ − x₁). Parallel lines have equal slopes; perpendicular lines have m₁m₂ = −1.
- Line through (x₁, y₁) with slope m: y − y₁ = m(x − x₁). The form y = mx + c has intercept c on the y-axis.
- Area of a triangle with vertices: ½|x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|.
Worked example
Find the equation of the line through (1, 2) and (4, 8). Slope = (8 − 2)/(4 − 1) = 6/3 = 2. Using point-slope form: y − 2 = 2(x − 1), so y = 2x, or 2x − y = 0. Check with (4, 8): 2(4) − 8 = 0.
Traps
Subtract coordinates in the same order in both numerator and denominator. A vertical line has undefined slope, so y = mx + c cannot represent it. In the section formula, the ratio is often given as “divides in 2:3”; do not reverse the weights. The area formula already includes ½ - dropping it doubles the answer.