In the figure,$BC$ is a diameter of the circle and $\angle BAO = 60^{\circ}$. Then $\angle ADC$ is equal to: (in $^{\circ}$)

$O$ is the circumcentre of the triangle $ABC$ and $D$ is the mid-point of the base $BC$. Prove that $\angle BOD = \angle A$.

State whether each of the following statements is true or false:
$(1)$ $A$ line segment joining any two points of a circle is a diameter of the circle.
$(2)$ For any circle, $\text{diameter} = 2 \times \text{radius}$.
$(3)$ In a circle with radius $14 \text{ cm}$, the length of a chord can be $32 \text{ cm}$.

$AB$ 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 $CD$.

In a circle with centre $P$,$AB$ is a chord. If the length of a radius is $17\,cm$ and $AB = 30\,cm$,then find the distance of $AB$ from the centre $P$. (in $,cm$)

$AB$ and $AC$ are two chords of a circle of radius $r$ such that $AB = 2 AC$. If $p$ and $q$ are the distances of $AB$ and $AC$ from the centre,prove that $4 q^{2} = p^{2} + 3 r^{2}$.