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.
If B is a different point than F, then a is not 0 and
Since b and f are distances, hence positive, if follows that
if B is any other point on the line than F.