Show that of all line segments drawn from a given point not on it, the perpendicular line segment is the shortest.
Let's suppose that we have line . A is any point from where we draw perpendicular AB to line . Suppose there is any other random point C present on the line . Join A and C.
So, we have right angled at B.
We need to prove that AB is shorter than AC.
(AB is perpendicular to line ) (1)
(Angle sum property of triangle)
Putting (1) in the above equation, we get
It means that is the largest angle in .
(In any triangle, the side opposite to the larger angle is longer.)
Similarly, we can choose any random point present on line and can complete a triangle in the same way. AB would be shortest of all the line segments. We can prove this using the same method.