Coordinate Geometry
What this domain tests
Coordinate Geometry tests distance, midpoint and section formulas, area of a triangle from coordinates, and collinearity. AP SSC expects exact substitution and careful sign handling.
Core rules
- Distance between (x₁,y₁) and (x₂,y₂): √[(x₂−x₁)² + (y₂−y₁)²].
- Midpoint of a segment: ((x₁+x₂)/2, (y₁+y₂)/2).
- Section formula for a point dividing the join of A(x₁,y₁) and B(x₂,y₂) internally in the ratio m₁:m₂: ((m₁x₂+m₂x₁)/(m₁+m₂), (m₁y₂+m₂y₁)/(m₁+m₂)).
- Area of a triangle: ½|x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|.
- Three points are collinear exactly when the triangle area is 0.
- Centroid: ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3).
Worked example
Show that (1,2), (4,6) and (7,10) are collinear. Area = ½|1(6−10) + 4(10−2) + 7(2−6)| = ½|−4 + 32 − 28| = ½(0) = 0. Since the area is zero, the three points lie on one straight line.
Traps
- Dropping the absolute value and reporting a negative area.
- Reversing the ratio in the section formula; the ratio corresponds to the two parts, not to the two endpoints.
- Forgetting to square both differences under the distance root.
- Reporting distance² as distance.
- Using the centroid formula when the midpoint or section formula is required.