Coordinate Geometry
Coordinate Geometry tests distance, section formula, midpoint and simple collinearity. The exam expects clean substitution and correct signs.
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 a point dividing the join of A(x1,y1) and B(x2,y2) in ratio m:n internally: ((mx2+nx1)/(m+n),(my2+ny1)/(m+n)).
- A point on the x-axis has y=0; on the y-axis x=0.
- Three points are collinear if the area of the triangle formed is zero or if distances satisfy the triangle equality.
Worked example
Find the distance AB for A(2,3), B(6,6). AB=√[(6-2)²+(6-3)²]=√(16+9)=√25=5. The midpoint is ((2+6)/2,(3+6)/2)=(4,4.5).
Traps
Squaring removes sign, but the subtraction order must be consistent; (x1-x2)² is safe. In the section formula, m:n is the ratio from the first point to the second-reversing it changes the point. Do not add coordinates before dividing by 2 for a midpoint. For a point on an axis, set the other coordinate to zero. Keep units if coordinates represent lengths.