Prove that the line joining the midpoint of a chord to the centre of the circle passes through the mid-point of the corresponding minor arc.

Two chords $AB$ and $CD$ of lengths $5 \mathrm{cm}$ and $11 \mathrm{cm}$ respectively of a circle are parallel to each other and are opposite side of its centre. If the distance between $AB$ and $CD$ is  $6 \mathrm{cm},$ if the radius of the circle is $\frac{k\sqrt{k}}{2} \mathrm{cm}$, then find the value of $k$.

In the following figure,$O$ is the centre of the circle. It $\angle APB=50°,$ find $\angle AOB$ and $\angle OAB$.

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If $O$ is the centre of the circle, find the value of $x$ in the following figure:

If $O$ is the centre of the circle, find the value of $x$ in the following figure.

If $O$ is the centre of the circle, find the value of $x$ in the given figure.

If $O$ is the centre of the circle, find the value of $x$ in the following figure.

If $O$ is the centre of the circle, find the value of $x$ in the given figure.

Find the value of $x$ in the given figure.