If a diameter of a circle bisects each of the two chords of a circle then prove that the chords are parallel.

In the adjoining figure, two circles with centers at $\mathrm{A}$ and $\mathrm{B}$ and of radii $5 \mathrm{cm} \mathrm{and} 3 \mathrm{cm}$ touch each other internally. If the perpendicular bisector of $AB$ meets the bigger circle in$\mathrm{P} \mathrm{and} \mathrm{Q}$ then the length of $\mathrm{PQ}$ is $A\sqrt{6}$. Find the value of $A$.

In the given figure, $\mathrm{AB}$ is a chord of a circle with center $\mathrm{O}$ and $\mathrm{AB}$ is produced to $\mathrm{C}$ such that  $\mathrm{BC}=\mathrm{OB}$ Also, $CO$ is joined and produced to meet the circle in $D.$ If $\angle ACD=y^{°}$ and $\angle \mathrm{AOD}=\mathrm{x}°$ prove that $\mathrm{x}=3\mathrm{y}$

In the adjoining figure, $\mathrm{O}$ is the center of a circle. If $\mathrm{AB}$ and $\mathrm{AC}$ are chords of the circle such that $\mathrm{AB}=\mathrm{AC}, \mathrm{OP}\perp \mathrm{AB}\text{ and }\mathrm{OQ}\perp \mathrm{AC}$. Prove that $\mathrm{PB}=\mathrm{QC}$.

In the adjoining figure, $\mathrm{BC}$ is a diameter of a circle with center  $\mathrm{O}$. If $\mathrm{AB}$ and $CD$ are two chords such that $\mathrm{AB}\Vert \mathrm{CD}$ prove that $\mathrm{AB}=\mathrm{CD}$.

An equilateral triangle of side $9 \mathrm{cm}$ is inscribed in a circle. If the radius of the circle is $k\sqrt{k} \mathrm{cm}$, then write the value of $k$.

In the adjoining figure, $\mathrm{AB}$ and $\mathrm{AC}$ are two equal chords of a circle with center  $\mathrm{O}$. Show that $\mathrm{O}$ lies on the bisector of $\angle \mathrm{BAC}$.

In the adjoining figure,$\mathrm{OPQR}$ is a square.$\mathrm{A}$ circle drawn with center $\mathrm{O}$ cuts the square in $\mathrm{X}$ and $\mathrm{Y}$ Prove that $\mathrm{QX}=\mathrm{QY}$