In a triangle $ABC$ with usual notations,if $\tan \left(\frac{B-C}{2}\right) = x \cot \frac{A}{2}$,then $x =$

With usual notations,in $\triangle ABC$,the lengths of two sides are $10 \text{ cm}$ and $9 \text{ cm}$ respectively. If angles $A, B, C$ are in $A$.$P$.,then the perimeter of $\triangle ABC$ is:

If in $\triangle ABC$,with usual notations,the angles are in $A$.$P$.,then $\frac{a}{c} \sin 2C + \frac{c}{a} \sin 2A =$

In a $\Delta ABC$,$\angle A = 30^\circ$ and $a = 8 \text{ cm}$,then the distance of the orthocentre from vertex $A$ is equal to (with usual notations).

In triangle $ABC$,if $a=7, b=8, \cos C=\frac{2}{7}$ and $C$ is an acute angle,then $c=$

Points $D$ and $E$ are taken on the side $BC$ of a triangle $ABC$ such that $BD = DE = EC$. If $\angle BAD = x$,$\angle DAE = y$,and $\angle EAC = z$,then the value of $\frac{\sin(x + y)\sin(y + z)}{\sin x \sin z}$ is: