Show that of all line segments drawn from a given point not on it, the perpendicular line segment is the shortest.
(Refer to Image)
Take a point P not on AB straight line, but some distance away from AB. We draw PC (the perpendicular on to AB). We draw another line from P to D.
In the triangle PCD, use the Pythagorean law for sides:
PC2+CD2=DP2
Since CD2 is positive and adds to PC2
∴PC2+CD2>PC2
or, DP2>PC2
So, DP>PC
So, PC the perpendicular distance from an external point to a line segment, is the shortest.