GeoProof — Distance & Midpoint explorer

Distance & Midpoint

On the coordinate plane, two points A = (x₁, y₁) and B = (x₂, y₂) have a distance and a midpoint given by simple formulas:

distance = √((x₂−x₁)² + (y₂−y₁)²)
midpoint = ( (x₁+x₂)/2 , (y₁+y₂)/2 )

Distance is the Pythagorean theorem

The horizontal gap Δx = x₂−x₁ and the vertical gap Δy = y₂−y₁ are the two legs of a right triangle whose hypotenuse is the segment AB. So AB² = Δx² + Δy² — the distance formula is just the Pythagorean theorem in coordinates. Turn on "Right triangle" to see the legs.

Midpoint is the average

The midpoint sits halfway along AB, so each of its coordinates is simply the average of the corresponding coordinates of A and B.

Try it

Drag A and B (they snap to whole-number grid points). A 3-4-5 example: put A at (0,0) and B at (3,4) — the distance reads 5, and the midpoint is (1.5, 2).