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 $,cm$)

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.

In a cyclic quadrilateral $ABCD$,$\angle A = 2x - 10^{\circ}$ and $\angle C = 3x - 35^{\circ}$,then $\angle A =$ .......... (in $^{\circ}$)

$AB$ and $AC$ are two chords of a circle with centre $P$. If the bisector of $\angle BAC$ passes through the centre $P$,prove that $AB = AC$.

Bisectors of angles $A, B$ and $C$ of a triangle $ABC$ intersect its circumcircle at $D, E$ and $F$ respectively. Prove that the angles of the triangle $DEF$ are $90^{\circ}-\frac{1}{2} A, 90^{\circ}-\frac{1}{2} B$ and $90^{\circ}-\frac{1}{2} C$.

$A$ quadrilateral $ABCD$ is inscribed in a circle such that $AB$ is a diameter and $\angle ADC = 130^{\circ}$. Find $\angle BAC$.