In the figure given below, the diagonals of a cyclic quadrilateral $ABCD$ intersect in $P$ and the area of the triangle $APB$ is $24 {\mathrm{cm}}^2$. If $\mathrm{AB}=8 \mathrm{cm} \mathrm{and} \mathrm{CD}=5 \mathrm{cm}$, calculate the area of $△DPC$.

In the figure given below, $\mathrm{QPX}$ is the bisector of $\angle \mathrm{YXZ}$ of the triangle $\mathrm{XYZ}$. Prove that $\mathrm{XY}:\mathrm{XQ}=\mathrm{XP}:\mathrm{XZ}.$

In the figure (ii) given below chords $\mathrm{BA}$ and $\mathrm{DC}$ of a circle meet at $\mathrm{P}$. Prove that

(i) $\angle \mathrm{PAD}=\angle \mathrm{PCB}$

(ii) $\mathrm{PA}\times \mathrm{PB}=\mathrm{PC}\times \mathrm{PD}$.

If $O$ is the centre of the circle, find the value of $x$ in the following figures (using the given information):

If $O$ is the centre of the circle, find the value of $x$ in the following figures (using the given information):

If $O$ is the centre of the circle, find the value of $x$ in the following figures (using the given information):

In the adjoining figure, $O$ is the centre of the circle. If $\angle \mathrm{AOC}=150°$, find $\angle ABC$.

In the adjoining figure, $O$ is the centre of the circle. If $\angle \mathrm{AOC}=150°$, find $\angle ADC$.

In the figure given below, $AC$ is a diameter of the given circle and $\angle BCD=75°$. Calculate the size of: $\angle ABC$