If each side of a regular dodecagon is $1 \mathrm{cm},$ find the longest diagonal of the dodecagon.

In the following figure $12$ rhombi are joined in such a way that they have a common vertex, without overlapping each other, and every rhombus shares two of its sides with the adjacent rhombi, as shown below. Find the area of the following figure, if its perimeter is $24 \mathrm{cm}.$

In the given figure, a square inscribes a dodecagon and each side of the regular dodecagon is $1 \mathrm{cm}.$ The dodecagon consists of $12$ equilateral triangles and $12$ rhombi. What is the ratio of area of dodecagon and the area of square?

The following figure consists of two concentric dodecagons. Find the ratio of area of smaller (interior) dodecagon to that of larger (exterior) dodecagon.

In the given diagram $\mathrm{PQ}$ is parallel to $\mathrm{RS}.$ For $\mathrm{PQ}>\mathrm{OP},$ when $\angle \mathrm{POQ}$ and $\angle \mathrm{OPQ}$ have integral values, find the greatest possible value of $\mathrm{OPQ}.$

In the following figure, $\mathrm{BC}$ and $\mathrm{RP}$ are tangents to the same circle with centre $\mathrm{O}$ and its diameter $\mathrm{AB}.$ Points $\mathrm{R}$ and $\mathrm{S}$ are the points on the tangent $\mathrm{BC},$ such that $\mathrm{AS}$ passes through $\mathrm{Q},$ where $\mathrm{RP}$ touches the circle.

Which of the following is necessarily a correct relations?

In the concerned diagrams, chords $\mathrm{AB}$ and $\mathrm{CD}$ are parallel and radius $\mathrm{OR}$ bisects the chord $\mathrm{AB}$ at $\mathrm{P}$ and $\mathrm{CD}$ at $\mathrm{Q}.$ Radius $\mathrm{OC}$ when extended to a point $\mathrm{T}$ in order to meet the tangent $\mathrm{TR}$ it intersects the chord $\mathrm{AB}$ at S. If $\mathrm{OP}: \mathrm{OS}=8: 17,$ find $\mathrm{OC}: \mathrm{OT}$