Coordinate Geometry
Coordinate Geometry tests distance, midpoint, section formula, collinearity, and simple area problems on the Cartesian plane. Points are written as ordered pairs (x,y).
Core rules
- Distance between (x1,y1) and (x2,y2): sqrt((x2-x1)^2 + (y2-y1)^2).
- Midpoint: ((x1+x2)/2, (y1+y2)/2).
- Section formula for a point dividing the join of A and B internally in ratio m1:m2: ((m1x2+m2x1)/(m1+m2), (m1y2+m2y1)/(m1+m2)).
- Three points are collinear if the area of the triangle formed is zero, or if the sum of two distances equals the third.
- A point on the x-axis has y=0; a point on the y-axis has x=0.
Worked example
Find the distance between A(2,3) and B(6,6). Difference in x is 4 and in y is 3. Distance = sqrt(4^2+3^2)=sqrt(16+9)=sqrt(25)=5 units. The midpoint is ((2+6)/2,(3+6)/2)=(4,4.5), which lies halfway along the same segment.
Traps
Subtract coordinates in the same order; reversing one pair without the other changes signs. The section formula is not the midpoint formula unless the ratio is 1:1. Do not forget to take the positive square root for distance. In collinearity, zero area is a condition, not the area itself.