Write True or False and justify your answer.
Two chords of a circle of lengths $10 \, cm$ and $8 \, cm$ are at the distances $8.0 \, cm$ and $3.5 \, cm$,respectively,from the centre.

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.

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 $, cm$)

Prove that two circles cannot intersect at more than two points.

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

$AB$ and $CD$ are two parallel chords of a circle with centre $P$. Also,the centre $P$ lies between the chords $AB$ and $CD$. If $AB = 20\,cm$,$CD = 48\,cm$,and the radius of the circle is $26\,cm$,find the distance between $AB$ and $CD$.