In $\Delta XYZ$,$m \angle Y = 90^{\circ}$ and $XY : XZ = 15 : 17$. If $YZ = 4$,find the perimeter of $\Delta XYZ$.

In $\Delta XYZ$,$m\angle Y = 90^{\circ}$ and $\overline{YP}$ is a median. If $YP = 6$,then $XZ = \ldots$

In $\Delta PQR$,$m\angle Q = 90^{\circ}$ and $\overline{QM}$ is an altitude. If $PM = x$ and $RM = y$,find the lengths of $\overline{PQ}$,$\overline{QR}$,$\overline{PR}$,and $\overline{QM}$ in terms of $x$ and $y$.

In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $\overline{BD}$ is an altitude to the hypotenuse $AC$. If $AD = 9$ and $CD = 27$,find $AB$.

In $\Delta ABC$,$D \in \overline{AB}$,$E \in \overline{AC}$ and $\overline{DE} \parallel \overline{BC}$. $F$ is a point on $\overline{AD}$ such that $\overline{EF} \parallel \overline{CD}$. Prove that $AD^2 = AB \times AF$.

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 $\ldots$