State whether the following statements are True or False. Give reasons for your answers.
$(i)$ The line segment joining the centre to any point on the circle is a radius of the circle.
$(ii)$ $A$ circle has only a finite number of equal chords.
$(iii)$ If a circle is divided into three equal arcs,each is a major arc.
$(iv)$ $A$ chord of a circle,which is twice as long as its radius,is a diameter of the circle.
$(v)$ $A$ sector is the region between the chord and its corresponding arc.
$(vi)$ $A$ circle is a plane figure.

(A-D) $(i)$ True. The definition of a radius is the line segment joining the centre to any point on the circle.
$(ii)$ False. $A$ circle can have an infinite number of equal chords of any given length.
$(iii)$ False. If a circle is divided into three equal arcs,each arc measures $120^{\circ}$,which is less than a semicircle $(180^{\circ})$,so each is a minor arc.
$(iv)$ True. By definition,the diameter is the longest chord of a circle and its length is equal to $2 \times \text{radius}$.
$(v)$ False. The region between a chord and its corresponding arc is called a segment,whereas a sector is the region between an arc and the two radii joining the centre to the endpoints of the arc.
$(vi)$ True. $A$ circle is a collection of all points in a plane that are at a fixed distance from a fixed point (the centre).