If the sides of a triangle are $13, 14, 15$,then the radius of its incircle is

Corresponding to a triangle $ABC$,match the items given in List-$I$ with the items given in List-$II$.

List-$I$	List-$II$
$(A)$ $rr_2 = r_1r_3$	$(I)$ $\angle A = 90^{\circ}$
$(B)$ $r_1 + r_2 = r_3 - r$	$(II)$ $b^2 = c^2 + a^2$
$(C)$ $r_1 = r + 2R$	$(III)$ $\angle C = 90^{\circ}$
	$(IV)$ $\angle B = 120^{\circ}$

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:

The circumradius of a triangle whose sides are $10 \ units$,$8 \ units$,and $6 \ units$ is (in $units$)

In a $\triangle ABC$,if $\frac{2 r_2 r_3}{r_2-r_1}=r_3-r_1$,then $\frac{r_1(r_2+r_3)}{\sqrt{r_1 r_2+r_2 r_3+r_3 r_1}} = $

Let $ABC$ be an equilateral triangle with side length $a$. Let $R$ and $r$ denote the radii of the circumcircle and the incircle of triangle $ABC$ respectively. Then,as a function of $a$,the ratio $\frac{R}{r}$