In this chapter we will prove that the perpendicular line is the shortest distance between a point and a line.
Given above is the line L and point P.
The distance between point P and line L is shown by two segment PM and PR.
PM is the perpendicular segment which intersect line L at 90 degree angle.
The shortest distance between point P and line M is shown by perpendicular line PM.
Consider triangle PMR.
∠PMR = 90 degree (as PM is perpendicular line )
We know that in a triangle, if one of the angle is right angle ( 90 degree ) then the other two angle would definitely be acute angle.
This means that the other two angles are less than ∠PMR .
∠PRM < ∠PMR
In a triangle, the side opposite to larger angle is greater.
Hence, side PM < PR.
So we proved that perpendicular line PM is the shorter side.