The angles of a triangle are in the ratio $2:3:7$ and the radius of the circumscribed circle is $10 \text{ cm}$. The length of the smallest side is (in $\text{ cm}$)

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

In $\triangle ABC$,if $\frac{1}{b+c} + \frac{1}{c+a} = \frac{3}{a+b+c}$,then $\angle C$ is equal to: (in $^{\circ}$)

Let $ABC$ be a triangle such that $AB=4, BC=5$ and $CA=6$. Choose points $D, E, F$ on $AB, BC, CA$ respectively,such that $AD=2, BE=2, CF=2$. Then find the ratio of the area of $\triangle DEF$ to the area of $\triangle ABC$.

In triangle $ABC$,if $BC$ is the hypotenuse,then $r_2 + r_3 =$

In a $\Delta ABC,$ if $(\sin A + \sin B + \sin C)(\sin A + \sin B - \sin C) = 3\sin A\sin B,$ then the angle $C$ is equal to