Out of two concentric circles,the radius of the outer circle is $5 \, cm$ and the chord $AC$ of length $8 \, cm$ is a tangent to the inner circle. Find the radius of the inner circle (in $cm$).

In $Fig.$,$AB$ and $CD$ are common tangents to two circles of unequal radii. Prove that $AB = CD$.

$P$ 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 $OP$,if $r = 12$ and $XP = 5$.

Let $P$ be the centre of a circle and $AB$ be a line in the same plane. If $Q$ is the foot of the perpendicular drawn from $P$ to the line $AB$,and $Q$ lies in the interior of the circle,then the line $AB$ ..... .

In $\Delta ABC$,$\angle B = 90^{\circ}$. The radius of the incircle touching all three sides of the triangle is $\ldots \ldots \ldots \ldots$

$A$ tangent $\stackrel{\leftrightarrow}{AB}$ of $\odot(P, r)$ touches the circle at $Q$. If a perpendicular is drawn from $P$ onto $AB$,then the foot of the perpendicular is ....