A circle has a centre $\mathrm{O}$ and radii $\mathrm{OQ}$ and $\mathrm{OR}$. Two tangents, $\mathrm{PQ}$ and $\mathrm{PR}$, are drawn from an external point $\mathrm{P}$.

In addition to the above information, which of these must also be known to conclude that the quadrilateral $\mathrm{PQOR}$ is a square?

i) $\mathrm{OQ}$ and $\mathrm{OR}$ are at an angle of $90°$.

ii) The tangents meet at an angle of $90°$.

Raghav drew the following figure on a board where a circle is inscribed in a quadrilateral.

(Note: The figure is not to scale)

Then he wrote the following relationships.

i) $\mathrm{ZW}+\mathrm{WX}=\mathrm{XY}+\mathrm{YZ}$

ii) $\mathrm{ZY}+\mathrm{WX}=\mathrm{ZW}+\mathrm{YX}$

Which of the above relationships is/are DEFINITELY true?

In the figure below, $∆\mathrm{PXY}$ is formed using three tangents to a circle centred at $\mathrm{O}$.

(Note: The figure is not to scale)

Based on the construction, the sum of the tangents $\mathrm{PA}$ and $\mathrm{PB}$ is_____the perimeter of $∆\mathrm{PXY}$.

Here is a circle with centre $O$.

Manu wants to draw a tangent $RS$  to the circle.
What is the number of points at which the line $RS$ will meet the circle?

$PQ$ is a chord of length $16 cm$ of a circle of radius $10 cm.$ The tangents at $P$ and $Q$ intersect at a point $T.$ as shown in the figure.

Find the length of $TP.$

In figure, $AT$ is a tangent to the circle with centre $0$ such that $OT=4 cm$ and $\angle OTA=30°$. Then, $AT$ is equal to

Tangents $PA$ and $PB$ inclined at an angle $60°$ as shown in the figure, the ratio of lengths of $OA, OP$ and $AP$ is