In $\triangle ABC$,if $a: b: c = 4: 5: 6$,then the ratio between the circumradius and the inradius is

If $\triangle ABC$ is right-angled at $A$, then $r_2+r_3$ is equal to

In a $\triangle ABC$,$D, E$,and $F$ are the points of contact of the incircle with the sides $AB, BC$,and $CA$ respectively,such that $AD=\alpha, BE=\beta$,and $CF=\gamma$. Then $\frac{\alpha \beta \gamma}{\alpha+\beta+\gamma} = $

If the roots of the equation $x^3-11x^2+36x-36=0$ are the ex-radii of a triangle $ABC$, then the perimeter of the triangle $ABC$ is

In $\Delta ABC,$ the lengths of sides $AC$ and $AB$ are $12 \, cm$ and $5 \, cm,$ respectively. If the area of $\Delta ABC$ is $30 \, cm^{2}$ and $R$ and $r$ are respectively the radii of the circumcircle and incircle of $\Delta ABC,$ then the value of $2R + r$ (in $cm$) is equal to ....... .

The perimeter of $\triangle ABC$ is $36 \text{ cm}$ and its inradius is $8 \text{ cm}$. Then,the area of the triangle is (in $\text{ cm}^2$)