The perpendicular from the centre of a circle to a chord ........... the chord.

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.

Prove that if the angles subtended by the chords of a circle (or of congruent circles) at the centre (or the corresponding centres) are equal,then the chords are equal.

$ABCD$ is a cyclic quadrilateral such that $AB$ is a diameter of the circle circumscribing it and $\angle ADC = 140^{\circ}$,then $\angle BAC$ is equal to (in $^{\circ}$)

$A$ chord of a circle is equal to its radius. Find the angle subtended by this chord at a point in the major segment.

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 $RS = \dots \, cm$.