Coordinate Geometry: Distance, Midpoint and Gradient | Form 2 Mathematics

Coordinate Geometry: Distance, Midpoint and Gradient

Coordinates allow geometric relationships to be calculated from ordered pairs.

Formulas

For A(x₁,y₁) and B(x₂,y₂), distance AB = √[(x₂−x₁)² + (y₂−y₁)²]. The midpoint is ((x₁+x₂)/2, (y₁+y₂)/2). Gradient m = (y₂−y₁)/(x₂−x₁), provided x₂ ≠ x₁.

Worked examples

1. For A(1,2) and B(4,6), distance = √(3²+4²) = 5.
2. The midpoint of (2,5) and (8,1) is (5,3).
3. The gradient through (1,2) and (5,10) is 8/4 = 2.

Practice

1. Find the distance between (0,0) and (6,8).
2. Find the midpoint of (−3,4) and (5,−2).
3. Find the gradient through (2,7) and (6,−1).