Mathematics

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


SOLUTION

(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:
$$PC^2+CD^2=DP^2$$
Since $$CD^2$$ is positive and adds to $$PC^2$$
$$\therefore PC^2+CD^2 >PC^2$$
or, $$DP^2 >PC^2$$
So, $$DP >PC$$
So, $$PC$$ the perpendicular distance from an external point to a line segment, is the shortest.
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Subjective Medium Published on 09th 09, 2020
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