In a triangle $ABC,$ $a:b:c = 4:5:6$. The ratio of the radius of the circumcircle to that of the incircle is

In triangle $ABC$,if $a=6, b=8$ and $c=10$,then $\frac{2 r_2 r_3}{r r_1} = $

If $r$ is the inradius,$\Delta$ is the area of $\triangle ABC$,and $s$ is the semi-perimeter,then which of the following is true?

In a $\triangle ABC$,let $a, b, c, s, r, R, I, S, r_1, r_2, r_3$ stand for their usual meanings. Match the items of List-$I$ with those of List-$II$.

List-$I$	List-$II$
$A. \tan \frac{A}{2} = \frac{r}{s-a}$	$I. (AI) \left( \frac{\sqrt{(s-b)(s-c)}}{bc} \right)$
$B. r$	$II. R^2$
$C. (SI)^2 + 2Rr$	$III. (4R + r + \sqrt{2}s)(4R + r - \sqrt{2}s)$
$D. r_1^2 + r_2^2 + r_3^2$	$IV. \frac{Rr}{S}$
	$V. \frac{(s-b)(s-c)}{\Delta}$

The correct match is:

The radius of the incircle of a triangle $PQR$ with vertices $P(0, 0)$,$Q(3, 0)$,and $R(0, 4)$ is:

$AD, BE$ and $CF$ are the perpendiculars from the vertices of a $\Delta ABC$ to the opposite sides. The ratio of the perimeter of $\Delta DEF$ to the perimeter of $\Delta ABC$ is: (where $r$ is the inradius and $R$ is the circumradius of $\Delta ABC$)