Question

Show that of all line segments drawn from a given point not on it, the perpendicular line segment is the shortest.

Answer 1

(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:

$P{C}^2+C{D}^2=D{P}^2$

Since $C{D}^2$ is positive and adds to $P{C}^2$

$\therefore P{C}^2+C{D}^2>P{C}^2$

or, $D{P}^2>P{C}^2$

So, $DP>PC$

So, $PC$ the perpendicular distance from an external point to a line segment, is the shortest.