In a right-angled $\triangle ABC$, the measures of the angles are in an Arithmetic Progression ($A$.$P$.). If its smallest side is $4 \text{ units}$, then the area of $\triangle ABC$ is:

Which of the following conditions allows for the existence of a triangle $ABC$?

Let $a, b$ and $c$ denote the lengths of sides $BC, CA$ and $AB$ of $\triangle ABC$. In $\triangle ABC$,$\angle BAC = 30^{\circ}$ and $\angle ABC = 60^{\circ}$. Then $a: b: c$ is

In a $\Delta ABC,$ if $\frac{b + c}{11} = \frac{c + a}{12} = \frac{a + b}{13},$ then $\cos C = $

In a triangle $PQR$,let $\angle PQR = 30^{\circ}$ and the sides $PQ$ and $QR$ have lengths $10\sqrt{3}$ and $10$,respectively. Then,which of the following statement$(s)$ is (are) $TRUE$?
$(A)$ $\angle QPR = 45^{\circ}$
$(B)$ The area of the triangle $PQR$ is $25\sqrt{3}$ and $\angle QRP = 120^{\circ}$
$(C)$ The radius of the incircle of the triangle $PQR$ is $10\sqrt{3} - 15$
$(D)$ The area of the circumcircle of the triangle $PQR$ is $100\pi$

In $\triangle ABC$,if $r_1 = 2r_2 = 3r_3$,then