In the figure,if $OA = 5 \text{ cm}$,$AB = 8 \text{ cm}$,and $OD$ is perpendicular to $AB$,then $CD$ is equal to (in $\text{cm}$):

In a circle with centre $P$,$AB$ and $CD$ are equal chords. If $\angle APB = 100^{\circ}$,then find $\angle PCD$. (in $^{\circ}$)

In a circle with centre $P$,the length of a radius is $\sqrt{2} \ cm$. It is divided into two segments by chord $AB$ of length $2 \ cm$. If $C$ is a point on the major arc $AB$,then find the value of $\angle ACB$. (in $^{\circ}$)

In a circle with centre $P$,$AB$ is a diameter and $ABCD$ is a cyclic quadrilateral. If $\angle ADC = 150^{\circ}$,then find $\angle BAC$. (in $^{\circ}$)

In the figure,$BC$ is a diameter of the circle and $\angle BAO = 60^{\circ}$. Then $\angle ADC$ is equal to: (in $^{\circ}$)

Prove that equal chords of a circle subtend equal angles at the centre.