The region between an arc and the two radii,joining the centre to the end points of the arc is called a ........... .

$AB$ and $CD$ are two parallel chords of a circle with centre $P$. Also,the centre $P$ lies between the chords $AB$ and $CD$. If $AB = 20\,cm$,$CD = 48\,cm$,and the radius of the circle is $26\,cm$,find the distance between $AB$ and $CD$.

In the same plane,two circles can have a maximum of ............ common chords.

If bisectors of opposite angles of a cyclic quadrilateral $ABCD$ intersect the circle circumscribing it at the points $P$ and $Q$,prove that $PQ$ is a diameter of the circle.

Prove that among all the chords of a circle passing through a given point inside the circle,the one that is perpendicular to the diameter passing through the point is the smallest.

$A$ circular park of radius $20\,m$ is situated in a colony. Three boys Ankur,Syed and David are sitting at equal distance on its boundary,each having a toy telephone in his hands to talk with each other. Find the length of the string of each phone.