Question

Question 6
Show that of all line segments drawn from a given point not on it, the perpendicular line segment is the shortest.

Answer 1

Let us take a line l and from point P (i.e. not on line l) we have drawn two line segments PN and PM. Let PN be perpendicular to line l and PM is drawn at some other angle.
In $\Delta$PNM
$\therefore \angle N={90}^{\circ }$
Now, $\angle P+\angle N+\angle M={180}^{\circ }$ (Angle sum property of a triangle)
Then $\angle P+\angle M={90}^{\circ }$
Clearly, M is an acute angle
$\therefore \angle M<\angle N$
$\Rightarrow$ PN < PM (side opposite to smaller angle is smaller)
Similarly by drawing different line segment from P to l, we can prove that PN is smaller as comparison to them. So, we may observe that of all line segments drawn from a given point not on it, the perpendicular line segment is the shortest.