SECONDARY MATHEMATICS · COORDINATE GEOMETRY REASONING
Midpoint and distance formulas are not isolated recipes. They encode two geometric ideas: halfway position and length.
Why the midpoint formula averages coordinates
For A(x₁,y₁) and B(x₂,y₂), the midpoint is ((x₁+x₂)/2,(y₁+y₂)/2). Its x-coordinate lies halfway between the two x-values, and its y-coordinate lies halfway between the two y-values.
For A(2,3) and B(8,11), the midpoint is (5,7). From A to M the change is (3,4); from M to B it is also (3,4). That equal displacement is the geometric reason M bisects AB.
Distance comes from Pythagoras
Between A(x₁,y₁) and B(x₂,y₂), horizontal change is Δx=x₂−x₁ and vertical change is Δy=y₂−y₁. These form perpendicular legs of a right triangle, so distance AB=√((Δx)²+(Δy)²).
For A(1,2) and B(4,6), Δx=3 and Δy=4, giving distance 5.
Distance is independent of direction
From A to B the changes may be (3,4); from B to A they are (−3,−4). Squaring gives the same distance. Length has no sign even though displacement components do.
Midpoints can prove diagonals bisect each other
For a quadrilateral, if the midpoint of diagonal AC equals the midpoint of diagonal BD, then the diagonals bisect each other. Under the usual Euclidean geometry result, this supports proving the quadrilateral is a parallelogram.
Example: A(0,0), B(4,1), C(6,5), D(2,4). Midpoint AC=(3,2.5) and midpoint BD=(3,2.5). The diagonals share the same midpoint.
Distances can prove equal sides
If AB and BC have equal calculated lengths, that equality can support an isosceles-triangle argument. Squared distances can sometimes be compared without taking square roots: if AB²=BC² and lengths are nonnegative, then AB=BC.
Use squared distance strategically
Suppose one length gives √52 and another also gives √52. It may be cleaner to show both squared lengths equal 52. The geometry depends on equality of lengths; taking identical square roots adds no new information.
Worked checks
A. Midpoint of (−2,5) and (6,−1) is (2,2).
B. Distance between (−1,2) and (5,10) is √(6²+8²)=10.
C. If two diagonals have the same midpoint, investigate parallelogram structure.
D. If two sides have equal squared distances, their lengths are equal.
Continue
Continue from Coordinates as Geometry and Gradient, Parallel and Perpendicular Lines into coordinate proof. Return to the Secondary Mathematics Master Index.