In $\Delta PQR$,$\angle Q$ is a right angle. If $PQ = 8$ and $QR = 15$,then the radius of a circle touching all the three sides of $\Delta PQR$ is $\ldots \ldots.$

In the following figure,if $AB = 10$,then $AC = \ldots$

If an isosceles triangle $ABC$, in which $AB = AC = 6\, cm$, is inscribed in a circle of radius $9\, cm$, find the area of the triangle in $cm^{2}$.

$P$ 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 $m \angle XPO$,if $m \angle XOY = 100^\circ$. (in $^\circ$)

If $a, b, c$ are the sides of a right triangle where $c$ is the hypotenuse,prove that the radius $r$ of the incircle of the triangle is given by $r = \frac{a + b - c}{2}$.

$AB$ is a diameter of a circle. $l_{1}$ and $l_{2}$ are tangents to the circle drawn at points $A$ and $B$ respectively. Then,which of the following is true?