If the perimeter of the triangle $ABC$ is $50 \text{ cm}$,then $b \cos^2 \frac{C}{2} + c \cos^2 \frac{B}{2} =$

In triangle $ABC$,if $\frac{b+c}{9}=\frac{c+a}{10}=\frac{a+b}{11}$,then $\frac{\cos A+\cos B}{\cos C}=$

In any triangle $ABC$,if $a: b: c = 2: 3: 4$,then $R: r =$

In a $\Delta ABC$,evaluate $\frac{b \sin(C - A)}{c^2 - a^2} + \frac{c \sin(A - B)}{a^2 - b^2}$.

In a triangle $ABC$,$m \angle A, m \angle B, m \angle C$ are in $A$.$P$. and the lengths of the two larger sides are $10$ units and $9$ units respectively. Then the length (in units) of the third side is:

The inradius of $\Delta ABC$ is $3$ units and the exradius opposite to vertex $A$ is $4$ units. Then,the length of the altitude from vertex $A$ is ............. $units$.