Two circles of radii $5\, cm$ and $3\, cm$ intersect at two points and the distance between their centres is $4\, cm.$ Find the length of the common chord. (in $, cm$)

Suppose you are given a circle. Give a construction to find its centre.

$A$ chord of a circle is equal to the radius of the circle. Find the angle subtended by the chord at a point on the minor arc and also at a point on the major arc.

Two circles intersect at two points $B$ and $C$. Through $B$,two line segments $ABD$ and $PBQ$ are drawn to intersect the circles at $A, D$ and $P, Q$ respectively (see figure). Prove that $\angle ACP = \angle QCD$.

Recall that two circles are congruent if they have the same radii. Prove that equal chords of congruent circles subtend equal angles at their centres.

$ABC$ and $ADC$ are two right triangles with a common hypotenuse $AC$. Prove that $\angle CAD = \angle CBD$.