The length of the common chord of two intersecting circles is $30$ cm. If the diameters of these two circles be $50$ cm and $34$ cm, calculate the distance between their centres.

The line joining the mid-points of two chords of a circle passes through its centre. Prove that the chords are parallel.

In the following figure, the line $ABCD$ is perpendicular to $PQ$; where $P$ and $Q$ are the centres of the circles. Show that: $AB=CD$.

In the following figure, the line $ABCD$ is perpendicular to $PQ$; where $P$ and $Q$ are the centres of the circles. Show that: $AC=BD$.

$AB\text{ and }CD$ are two equal chords of a circle with centre $O$ which intersect each other at a right angle at point $P$. If $OM\perp AB\text{ and }ON\perp CD$, show that $OMPN$ is a square.

In the given figure, an equilateral triangle $ABC$ is inscribed in a circle with centre $O$. Find $\angle BOC$.

In the given figure, an equilateral triangle $ABC$ is inscribed in a circle with centre $O$. Find $\angle OBA$.

In the given figure, a square is inscribed in a circle with centre $0$. Find $\angle BOC$

In the given figure, a square is inscribed in a circle with centre $0$. Find $\angle OCB$