The line segment joining the centre and any point on the circle is called a .......... of the circle.

In a circle with centre $O$,the lengths of two chords $AB$ and $CD$ are $12 \, cm$ and $16 \, cm$ respectively. If the chord $AB$ is at a distance $8 \, cm$ from the centre,what is the distance of the chord $CD$ from the centre (in $, cm$)?

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 the figure,$AOB$ is a diameter of the circle and $C, D, E$ are any three points on the semi-circle. Find the value of $\angle ACD + \angle BED$.

$AB$ and $XY$ are two chords of a circle whose centre is $P$. $A$ line $l$ passing through the centre $P$ bisects chords $AB$ and $XY$ both. Prove that $AB \parallel XY$.

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