In a circle with centre $P$,$AB$ is a chord. If the length of a radius is $17\,cm$ and $AB = 30\,cm$,then find the distance of $AB$ from the centre $P$. (in $,cm$)

$O$ is the circumcentre of the triangle $ABC$ and $D$ is the mid-point of the base $BC$. Prove that $\angle BOD = \angle A$.

$AB$ and $CD$ are two parallel chords of a circle with centre $P$. Centre $P$ is not between the two chords $AB$ and $CD$. If $AB = 40\,cm$,$CD = 30\,cm$ and the radius of the circle is $25\,cm$,find the distance between $AB$ and $CD$. (in $,cm$)

In quadrilateral $ABCD$,if $\angle A : \angle B : \angle C : \angle D = 3 : 2 : 4 : 5$,then identify the type of quadrilateral $ABCD$.

In a circle with centre $O$,$PQ$ and $RS$ are chords. If $\angle POQ = 90^{\circ}$,$\angle ROS = 90^{\circ}$ and $PQ = 6\, cm$,then $RS = \dots \, cm$.

$A$ 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.