Distance and Midpoint

Summary

The distance formula is the Pythagorean theorem in coordinates. The midpoint formula averages coordinates. Together they define length and midpoints of segments in the plane and in space.

Prerequisites

Coordinate plane (and optionally ${\mathbb{R}}^3$ ). Hub: Analytic Geometry.

Object / Concept

For points $P$ and $Q$ , the distance $d(P,Q)$ is the Euclidean length of the segment $PQ$ . The midpoint $M$ is the point halfway from $P$ to $Q$ along that segment.

Coordinate System

Cartesian coordinates in ${\mathbb{R}}^2$ or ${\mathbb{R}}^3$ with the standard Euclidean metric.

Notation

Conditions / Assumptions

• Euclidean (Pythagorean) distance, not taxicab or other metrics.
• Real coordinates.

Equations

Plane

Space

Circle with center $C=(h,k)$ and radius $r>0$ : all points $P$ with $d(P,C)=r$ , i.e.

Worked Example

Points $A=(1,2)$ and $B=(4,6)$ :

Common Mistakes

• Omitting the square root when a true distance (not squared distance) is required.
• Subtracting coordinates in inconsistent order (order cancels after squaring for distance, but midpoint needs correct averages).
• Using 2D formulas for points given in 3D.

Connections

• Related: Completing the Square, Vectors and Dot Product, Conic Sections
• Next: Lines and Planes
• Linear algebra: norms via Orthogonality and Projections

References

Distance and midpoint formulas are standard in OpenStax Precalculus analytic geometry.^[1]