State whether each of the following statements is true or false:
$(1)$ $A$ circle divides the plane on which it lies into three parts.
$(2)$ $A$ point,whose distance from the centre of a circle is greater than its radius,lies in the interior of the circle.

$AD$ is a diameter of a circle and $AB$ is a chord. If $AD = 34 \, cm$ and $AB = 30 \, cm$,the distance of $AB$ from the centre of the circle is (in $cm$):

In the figure,$AB$ and $CD$ are two chords of a circle intersecting each other at point $E$. Prove that $\angle AEC = \frac{1}{2}$ (angle subtended by arc $CXA$ at the centre $+$ angle subtended by arc $DYB$ at the centre).

In the figure,two congruent circles have centers $O$ and $O'$. Arc $AXB$ subtends an angle of $75^{\circ}$ at the center $O$ and arc $A'YB'$ subtends an angle of $25^{\circ}$ at the center $O'$. Then the ratio of the lengths of arcs $AXB$ and $A'YB'$ is:

In a circle with centre $O$,two chords $PQ$ and $RS$ subtend equal angles at the centre. If $PQ = 12 \text{ cm}$,then $RS = \dots \text{ cm}$.

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 $32 \text{ cm}$.