In a circle with radius $17 \, cm$,the length of a chord is $30 \, cm$. Find the distance of the chord from the centre. (in $, cm$)

In cyclic quadrilateral $PQRS$,if $\angle PRQ = 52^{\circ}$,then find $\angle PSQ$. (in $^{\circ}$)

$ABCD$ is a quadrilateral such that $A$ is the centre of the circle passing through $B, C$ and $D$. Prove that $\angle CBD + \angle CDB = \frac{1}{2} \angle BAD$.

In the figure,$OD$ is perpendicular to chord $AB$ of a circle whose centre is $O.$ If $BC$ is a diameter,prove that $CA = 2OD$.

Write True or False and justify your answer in each of the following:
Two congruent circles with centres $O$ and $O^{\prime}$ intersect at two points $A$ and $B$. Then $\angle AOB = \angle AO^{\prime}B$.

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