Mathematics.

coordinate geometry

Distance and Midpoint Formulas

Analytic Geometry15 minDifficulty2 out of 10

You should know: coordinate plane

Overview

The distance formula computes the straight-line distance between two points in the coordinate plane, and the midpoint formula finds the point exactly halfway between them. Both are direct consequences of the Pythagorean theorem applied to the horizontal and vertical legs of a right triangle formed by the two points and their coordinate differences. Together they underlie nearly every other formula in analytic geometry — circle equations, conic sections, and vector magnitudes all reduce to the distance formula in disguise.

Intuition

Draw the horizontal segment from P₁ to the point (x₂, y₁) and the vertical segment up to P₂: these two legs, of lengths |x₂−x₁| and |y₂−y₁|, together with the segment P₁P₂ itself, form a right triangle. The Pythagorean theorem then says the straight-line distance is the hypotenuse, √((Δx)²+(Δy)²) — the distance formula is nothing more than the Pythagorean theorem written in coordinates. The midpoint formula is just as direct: averaging the x-coordinates finds the horizontal middle, and averaging the y-coordinates finds the vertical middle.

Formal Definition

Definition

For two points P₁ = (x₁, y₁) and P₂ = (x₂, y₂) in the coordinate plane:

d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}
Distance formula
M=(x1+x22, y1+y22)M = \left(\frac{x_1+x_2}{2},\ \frac{y_1+y_2}{2}\right)
Midpoint formula

Worked Examples

  1. Apply the distance formula.

    d=(4−1)2+(6−2)2=9+16=25=5d = \sqrt{(4-1)^2+(6-2)^2} = \sqrt{9+16} = \sqrt{25} = 5

  2. Apply the midpoint formula.

    M=(1+42,2+62)=(2.5, 4)M = \left(\frac{1+4}{2}, \frac{2+6}{2}\right) = (2.5,\ 4)

Answer: Distance = 5, Midpoint = (2.5, 4)

Practice Problems

Difficulty 2/10

Find the distance between (−3, 2) and (1, −1).

Difficulty 3/10

Find the midpoint of the segment joining (−4, 7) and (6, −3).

Difficulty 5/10

Two cell towers are located at (0, 0) and (30, 40) (km). A relay station is to be built at the midpoint. Find the relay's coordinates and the distance between the towers.

Quiz

The distance formula is a direct application of:
The distance between (1,2) and (4,6) is:
The midpoint of a segment is found by:

Summary

  • Distance formula d = √((x₂−x₁)² + (y₂−y₁)²) comes directly from the Pythagorean theorem.
  • Midpoint formula M = ((x₁+x₂)/2, (y₁+y₂)/2) averages the coordinates of the two endpoints.
  • These two formulas are the building blocks for circle equations, conic sections, and vector magnitude in the coordinate plane.

References

  1. WebsiteWikipedia — Euclidean distance