If $A = 30^\circ, c = 7\sqrt{3}$ and $C = 90^\circ$ in $\Delta ABC$,then $a = $

If the angles of a triangle are in the ratio $1: 1: 4$,then the ratio of the perimeter of the triangle to its largest side is

For the triangle $ABC$,with usual notations,if the angles $A, B, C$ are in $A.P.$ and $m \angle A = 30^{\circ}, c = 3$,then the values of $a$ and $b$ are respectively

In $\triangle ABC$,$a^2 \sin 2B + b^2 \sin 2A =$

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 $\text{ units}$)

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