The lengths of two parallel chords of a circle are $6\, cm$ and $8\, cm$. If the smaller chord is at a distance of $4\, cm$ from the centre,what is the distance of the other chord from the centre (in $, cm$)?

$AC$ and $BD$ are chords of a circle which bisect each other. Prove that $AC$ and $BD$ are diameters.

If two intersecting chords of a circle make equal angles with the diameter passing through their point of intersection,prove that the chords are equal.

Prove that the line of centres of two intersecting circles subtends equal angles at the two points of intersection.

$A$ chord of a circle is equal to the radius of the circle. Find the angle subtended by the chord at a point on the minor arc and also at a point on the major arc.

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.