$P$ is a point in the exterior of $\odot(O, r)$ and the tangents from $P$ to the circle touch the circle at $X$ and $Y$. Find $r$,if $OP = 30$ and $PY = 24$.

In quadrilateral $ABCD$,$m \angle D = 90^\circ$. $A$ circle with center $O$ and radius $r$ touches its sides $AB, BC, CD,$ and $DA$ at points $P, Q, R,$ and $S$ respectively. If $BC = 38, CD = 25,$ and $BP = 25,$ find the radius $r$ of the circle.

Write 'True' or 'False' and give reasons for your answer.
If a chord $AB$ subtends an angle of $60^{\circ}$ at the centre of a circle,then the angle between the tangents at $A$ and $B$ is also $60^{\circ}$.

If $AB$ is a chord of a circle with centre $O$,$AOC$ is a diameter and $AT$ is the tangent at $A$ as shown in the figure. Prove that $\angle BAT = \angle ACB$.

$A$ is a point at a distance $13\, cm$ from the centre $O$ of a circle of radius $5 \,cm$. $AP$ and $AQ$ are the tangents to the circle at $P$ and $Q$. If a tangent $BC$ is drawn at a point $R$ lying on the minor arc $PQ$ to intersect $AP$ at $B$ and $AQ$ at $C$,find the perimeter of the $\triangle ABC$. (in $cm$)

$\overline{AB}$ is a chord of a circle with centre $O$. Line $l$ touches the circle at $B$. The foot of the perpendicular from $A$ to $l$ is $D$. Prove that $\angle BAO \cong \angle BAD$.