Prove that the shortest of all chords passing through a poin

Prove that the shortest of all chords, passing through a point A taken in the interior of a given circle, is the one which is perpendicular to the diameter drawn through A.

Solution

Let the chord passing through A makes an angle of X with the diameter drawn through A.

the length of the chord would therefore be equal to 2RcosX.

2RcosX is minimum when cosX = 0 or X = 90 degree

X = 90⁰ means chord is perpendicular to the diameter drawn through that point.