In $\Delta ABC$,$m \angle B = 90$ and $D$ is the midpoint of $\overline{AC}$. Then,$BD = \dots$

$\Delta ABC \sim \Delta XYZ$ for the correspondence $ABC \leftrightarrow XYZ$. If $AB = 12, XY = 8$ and the area of $\Delta ABC$ is $81$,find the area of $\Delta XYZ$.

In $\Delta PQR$ and $\Delta XYZ$,$\angle P \cong \angle X$ and $\angle Q \cong \angle Z$. If $PQ = 9$,$QR = 6$,$PR = 4.5$,and $XY = 7.5$,find $YZ$ and $XZ$.

In $\Delta PQR,$ the midpoints of the sides are $X, Y, Z.$ In $\Delta XYZ,$ the midpoints of the sides are $A, B, C.$ If the area of $\Delta PQR$ is $240,$ find the area of $\Delta XYZ$ and $\Delta ABC.$

In a triangle $PQR$,$N$ is a point on $PR$ such that $QN \perp PR$. If $PN \cdot NR = QN^2$,prove that $\angle PQR = 90^{\circ}$.

In $\Delta PQR$,$m \angle Q = 90^\circ$. If $PQ = 5$ and $PR = 13$,find the area of $\Delta PQR$.