The ratios of the sides of a triangle $ABC$ are $5:12:13$ and its area is $270$. Then the sides of the triangle are:

In $\triangle ABC$,if $r=3$ and $R=5$,then $\frac{1}{ab} + \frac{1}{bc} + \frac{1}{ca} = $

In $\triangle ABC$,if $r_1: r_2 = 7: 8$ and $r_1: r_3 = 3: 4$,then $a: b: c =$

Assertion $(A)$: In $\triangle ABC$,if $r=6, r_2=36, R=15$,then $c^2+a^2=b^2$.
Reason $(R)$: In $\triangle ABC$,if $r:R:r_2=1:2.5:6$,then $B=90^{\circ}$.
The correct option among the following is:

In a $\triangle ABC$,if $a+c=5b$,then $\cot \frac{A}{2} \cot \frac{C}{2} =$

If the area of a triangle $ABC$ is $4\sqrt{5} \text{ sq. units}$, the length of the side $CA$ is $6 \text{ units}$, and $\tan \frac{B}{2} = \frac{\sqrt{5}}{4}$, then the length of its smallest side is: (in $\text{ units}$)