Coordinate Geometry
What the domain tests
Coordinate geometry checks plotting points, identifying quadrants, and using the distance, midpoint, section and area formulas on the Cartesian plane.
Core rules and concepts
- A point is written (x, y); x is the horizontal coordinate and y is the vertical coordinate.
- Distance between (x₁, y₁) and (x₂, y₂) is √[(x₂ − x₁)² + (y₂ − y₁)²].
- Midpoint of a segment is ((x₁ + x₂)/2, (y₁ + y₂)/2).
- Section formula for dividing AB in ratio m:n is ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)).
- Area of a triangle with vertices (x₁, y₁), (x₂, y₂), (x₃, y₃) is ½|x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|. Collinear points give area 0.
Worked example
For A(−2, 3) and B(4, −5), the distance is √[(4 − (−2))² + (−5 − 3)²] = √(6² + (−8)²) = √100 = 10. The midpoint is ((−2 + 4)/2, (3 + (−5))/2) = (1, −1).
Traps that cost marks
Subtract coordinates in the same order; reversing one pair changes the sign inside the square, which is safe for distance but not for the section formula. Keep the absolute value in the area formula because area cannot be negative. Do not confuse the midpoint with a point that divides a segment in a ratio other than 1:1.