Question

Prove that the chord which passes through a point inside the circle is the smallest amongst all the chords and is perpendicular to the diameter passing through that point.

Answer 1

Given :
$AB$ is diameter passing through point $R$ inside circle of center $O$ and required chord is $PQ$ which is perpendicular to $AB,CD$ is other chord passing through $R$.
To Prove : Chord $PQ$ is the smallest.
Construction : Through $O$, draw $OM\perp CD$
Proof : In right angled $\Delta OMR$
Hypotenuse $OR$ is longest side of $\Delta OMR$
$OR>OM$
$OR$ is distance of chord $PQ$ from center $O$ and $OM$ is distance of chord $CD$ from center $O$. Since any one of two chords of a circle, the larger chord is nearer to the center. Thus, chord $PQ$ is the smallest