Coordinate Geometry: Distance, Section Formula, and Lines
Distance and section formulae
For points A(x₁,y₁), B(x₂,y₂), AB=√[(x₂−x₁)²+(y₂−y₁)²]. The midpoint is ((x₁+x₂)/2,(y₁+y₂)/2). The section formula for P dividing AB internally in m:n is ((mx₂+nx₁)/(m+n),(my₂+ny₁)/(m+n)). For external division, use m:n with a minus sign in the denominator.
Lines
Slope m=(y₂−y₁)/(x₂−x₁). A line y=mx+c has slope m; ax+by+c=0 has slope −a/b (b≠0). Parallel lines have equal slopes; perpendicular lines satisfy m₁m₂=−1. The angle θ between two lines satisfies tanθ=|(m₁−m₂)/(1+m₁m₂)|. Equation through (x₁,y₁) with slope m: y−y₁=m(x−x₁).
Worked example
Find the distance between A(2,3) and B(5,7): AB=√[(5−2)²+(7−3)²]=√(9+16)=5. The midpoint is ((2+5)/2,(3+7)/2)=(3.5,5). The slope of AB is (7−3)/(5−2)=4/3.
Traps
Subtracting coordinates in the wrong order inside squares is harmless, but reversing x and y in the section formula is not. A vertical line has undefined slope, so do not write m=0. For perpendicularity use the product −1, not equality. In the angle formula, take the absolute value and remember tan90° is undefined when 1+m₁m₂=0.