In a cyclic quadrilateral $\mathrm{ABCD}$, twice the measure of $\angle \mathrm{A}$ is thrice the measure of $\angle \mathrm{C}$. Then, the measure of $\angle \mathrm{C}$ is

Segment $\mathrm{XZ}$ is a diameter of a circle. Point $\mathrm{Y}$ lies in its interior. How many of the following statements are true?

$\left(i\right)$ It is not possible that $\angle \mathrm{XYZ}$ is an acute angle.

$\left(ii\right)$ $\angle \mathrm{XYZ}$ can't be a right angle.

$\left(iii\right)$ $\angle \mathrm{XYZ}$ is an obtuse angle.

$\left(iv\right)$ Can't make a definite statement for measure of $\angle \mathrm{XYZ}$.

Line $l$ touches a circle with centre $\mathrm{O}$ at point $\mathrm{P}$. If radius of the circle is $9\mathrm{cm}$, what is the distance between the points $\mathrm{O}$ and $\mathrm{P}$? Why?

In figure $\mathrm{M}$ is the centre of the circle and segment $\mathrm{KL}$ is a tangent segment. If $\mathrm{MK}=12,\mathrm{KL}=6\sqrt{3}$, then find radius of the circle.

In figure, $\mathrm{M}$ is the centre of the circle and segment $\mathrm{KL}$ is a tangent segment. If $\mathrm{MK}=12,\mathrm{KL}=6\sqrt{3}$, then find Measures of $\angle \mathrm{K}$ and $\angle \mathrm{M}$.

In figure, $\mathrm{O}$ is the centre of the circle. $\mathrm{AB}$ and $\mathrm{AC}$ are tangent segments. Radius of the circle is $\mathrm{r}$ and length of $\mathrm{AB}=\mathrm{r}$. Prove that $\mathrm{ABOC}$ is a square.