Two concentric circles are of diameters $30 \mathrm{c}\mathrm{m}$ and $18 \mathrm{c}\mathrm{m}.$ Find the length of the chord of the larger circle which touches the smaller circle.

In the given figure, the tangent at a point $\mathrm{C}$ of a circle and diameter $\mathrm{AB}$ when extended intersect at $\mathrm{P}. \mathrm{I}\mathrm{f} \angle \mathrm{P}\mathrm{C}\mathrm{A}=110°$, find $\angle \mathrm{C}\mathrm{B}\mathrm{A}$.

$\mathrm{AB}$ is a chord of a circle with centre $\mathrm{O}$, $\mathrm{AOC}$ is a diameter and $\mathrm{AT}$ is the tangent at $\mathrm{A}$ as shown in the figure.  Prove that $\mathbf{\angle }\mathrm{B}\mathrm{A}\mathrm{T}=\mathbf{\angle }\mathrm{A}\mathrm{C}\mathrm{B}.\mathrm{}$

In the given figure, a $△\mathrm{ABC}$ is drawn to circumscribe a circle of radius $4 \mathrm{cm}$ such that the segments $\mathrm{BD}$ and $\mathrm{CD}$ are of length $8 \mathrm{cm}$ and $6 \mathrm{cm}$ respectively. Find the lengths of sides $\mathrm{AB}$ and $\mathrm{AC}$, when area of $△\mathrm{ABC}$ is $84 {\mathrm{cm}}^2$.

In the given figure, $\mathrm{AB}$ is a diameter of a circle with centre $\mathrm{O}$ and $\mathrm{AT}$ is a tangent. If $\mathrm{\angle }\mathrm{A}\mathrm{O}\mathrm{Q}=58°,$ find $\mathrm{\angle }\mathrm{A}\mathrm{T}\mathrm{Q}.$

In the given figure, $\mathrm{O}\mathrm{Q}:\mathrm{P}\mathrm{Q}=3:4$ and perimeter of $\mathrm{\Delta }\mathrm{P}\mathrm{O}\mathrm{Q}=60 \mathrm{c}\mathrm{m}.$ Determine $\mathrm{PQ}, \mathrm{QR}$ and $\mathrm{OP}$.