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,if $AOB$ is a diameter of the circle and $AC = BC$,then $\angle CAB$ is equal to (in $^{\circ}$)

State whether each of the following statements is true or false:
$(1)$ The region between a chord and either of its arcs is called a sector.
$(2)$ The region between an arc and the two radii,joining the centre to the end points of the arc,is called a segment.

$AB$ is a chord of a circle with centre $P$. Point $C$ is a point other than $A$ and $B$ on the major arc $AB$. If $\angle ACB = 50^{\circ}$,then find $\angle APB$. (in $^{\circ}$)

Write True or False and justify your answer.
The angles subtended by a chord at any two points of a circle are equal.

If $BM$ and $CN$ are the altitudes drawn on the sides $AC$ and $AB$ of the triangle $ABC$,prove that the points $B, C, M$ and $N$ are concyclic.