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 $b \cos \theta = c - a$,(where $\theta$ is an acute angle),then $(c - a) \tan \theta =$

In $\triangle ABC$,if the median $AD$ drawn through $A$ is perpendicular to the side $AC$,then $3ca \cos A \cos C + 2a^2 =$

If $p_1, p_2, p_3$ are altitudes of a triangle $ABC$ from the vertices $A, B, C$ respectively,and $\Delta$ is the area of the triangle,then $p_1^{-1} + p_2^{-1} - p_3^{-1}$ is equal to

If the lengths of the sides of a triangle are $7 \, cm$,$4\sqrt{3} \, cm$,and $\sqrt{13} \, cm$,then the smallest angle is .....$^o$.

In a $\Delta ABC$,the ratio of angles $A : B : C = 3 : 5 : 4$. Then $a + b + c \sqrt{2}$ is equal to