1. The shortest distance from a point to a line is the perpendicular distance.

Let A be a point, let F be the point where the line through A which is perpendicular to the given line meets the given line, and let B be any other point on the line.

Since AF is perpendicular to BF, triangle ABF is a right triangle, so we can use the Pythagorean Theorem.

a2 + b2 = f2

If B is a different point than F, then a is not 0 and

b2 < f2

Since b and f are distances, hence positive, if follows that

b < f

if B is any other point on the line than F.