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 $RS = \dots \, cm$.

State whether each of the following statements is true or false:
$(1)$ For any two points on a circle,the line segment joining them is called a chord of the circle.
$(2)$ The length of a diameter of a circle is half the length of its radius.

Three girls Reshma,Salma and Mandip are playing a game by standing on a circle of radius $5 \, m$ drawn in a park. Reshma throws a ball to Salma,Salma to Mandip,Mandip to Reshma. If the distance between Reshma and Salma and between Salma and Mandip is $6 \, m$ each,what is the distance between Reshma and Mandip?

State whether the following statement is True or False and justify your answer: If $AOB$ is a diameter of a circle and $C$ is a point on the circle,then $AC^{2} + BC^{2} = AB^{2}$.

In the figure,$AB$ and $CD$ are two equal chords of a circle with center $O$. $OP$ and $OQ$ are perpendiculars on chords $AB$ and $CD$ respectively. If $\angle POQ = 150^{\circ}$,then $\angle APQ$ is equal to: (in $^{\circ}$)

$AB$ and $AC$ are two chords of a circle of radius $r$ such that $AB = 2 AC$. If $p$ and $q$ are the distances of $AB$ and $AC$ from the centre,prove that $4 q^{2} = p^{2} + 3 r^{2}$.