In a circle with centre $P$,$AB$ and $CD$ are chords. If $\angle APB = 80^{\circ}$,$\angle CPD = 50^{\circ}$ and $AB = 7\, cm$,then find the length of chord $CD$. (in $, cm$)

In a circle with centre $P$,$AB$ is a chord and point $C$ is a point other than $A$ and $B$ on the major arc $AB$. If $\angle ACB + \angle APB = 150^{\circ}$,then find $\angle APB$. (in $^{\circ}$)

$AB$ 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 $4 q^{2} = p^{2} + 3 r^{2}$.

Write True or False and justify your answer in each of the following:
If $A, B, C, D$ are four points such that $\angle BAC = 30^{\circ}$ and $\angle BDC = 60^{\circ}$, then $D$ is the centre of the circle through $A, B$ and $C$.

In a circle with centre $P$,the length of a diameter is $20 \, cm$ and $AB$ is a chord of a circle at a distance $6 \, cm$ from the centre,then $AB = $ ........ $cm$.

If a line is drawn parallel to the base of an isosceles triangle to intersect its equal sides,prove that the quadrilateral so formed is cyclic.