Question

prove that the shortest segment from a point to a line is the perpendicular segment .

Answer 1

One method uses the pythagorean theorem. If X and Y are two distinct points on some line ℓ, P is some point NOT on ℓ, and PX is perpendicular to ℓ, then triangle PXY is a right triangle, so the pythagorean theorem says that

(PX)2+(XY)2=(PY)2

and since (XY)2>0, we see that (PX)<(PY).

Since Y could be any other point on ℓ, this means that the shortest distance from P to ℓ is PX