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 $RS = \dots \, cm$.

The longest chord of a circle is a .......... of the circle.

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 $,cm$)

In the figure,$AOB$ is a diameter of the circle and $C, D, E$ are any three points on the semi-circle. Find the value of $\angle ACD + \angle BED$.

Write True or False and justify your answer in each of the following:
If $A, B, C, D$ are four points such that $\angle BAC = 30^{\circ}$ and $\angle BDC = 60^{\circ}$, then $D$ is the centre of the circle through $A, B$ and $C$.

The circumcentre of the triangle $ABC$ is $O$. Prove that $\angle OBC + \angle BAC = 90^{\circ}$.