The lengths of the diagonals of a rhombus are $16 \, cm$ and $12 \, cm$. Then,the length of the side of the rhombus is (in $cm$)

In $\Delta ABC$,$\overline{AD}$ is a median. If $AB^2 + AC^2 = 122$ and $AD = 6$,find $BC$.

In the figure,$O$ is a point in the interior of a triangle $ABC$,$OD \perp BC$,$OE \perp AC$,and $OF \perp AB$. Show that $OA^{2} + OB^{2} + OC^{2} - OD^{2} - OE^{2} - OF^{2} = AF^{2} + BD^{2} + CE^{2}$.

In $\triangle PQR$,$PD \perp QR$ such that $D$ lies on $QR$. If $PQ = a$,$PR = b$,$QD = c$,and $DR = d$,prove that $(a+b)(a-b) = (c+d)(c-d)$.

In $\Delta ABC$,$AB = 4$,$BC = 2\sqrt{3}$,and $AC = 2\sqrt{7}$. Then,in $\Delta ABC$,the length of the median on the longest side is $\ldots$

In rhombus $XYZW$,$XZ = 14$ and $YW = 48$. Then,$XY = \ldots \ldots \ldots$.