Two chords $AB$ and $CD$ of lengths $5\, cm$ and $11\, cm$ respectively of a circle are parallel to each other and are on opposite sides of its centre. If the distance between $AB$ and $CD$ is $6\, cm$,find the radius of the circle.

Let the vertex of an angle $\angle ABC$ be located outside a circle and let the sides of the angle intersect equal chords $AD$ and $CE$ with the circle. Prove that $\angle ABC$ is equal to half the difference of the angles subtended by the chords $DE$ and $AC$ at the centre.

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.

In the figure,$ABCD$ is a cyclic quadrilateral in which $AC$ and $BD$ are its diagonals. If $\angle DBC = 55^{\circ}$ and $\angle BAC = 45^{\circ}$,find $\angle BCD$. (in $^{\circ}$)

Given an arc of a circle,complete the circle.

Prove that a cyclic parallelogram is a rectangle.