In $\Delta XYZ$,$X-P-Y$,$X-Q-Z$ and $\overline{PQ} \parallel \overline{YZ}$. If $XY = 5.6$,$XP = 1.4$ and $XZ = 7.2$,find $XQ$.

$\Delta ABC \sim \Delta XYZ$ 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 $\Delta PQR$,$m\angle P = 90^{\circ}$ and $\overline{PM}$ is an altitude. If $PQ = \sqrt{20}$ and $QM = 4$,then $RM = \ldots \ldots .$

In $\square PQRS$,$\overline{PQ} \parallel \overline{RS}$ and $\overline{PR} \cap \overline{QS} = \{X\}$. If $XP = 2x + 4$,$XQ = 4x - 2$,$XR = x + 1$,and $XS = 4$,find the value of $x$.

Point $P$ lies in the interior of rectangle $ABCD$. Prove that $PA^2 + PC^2 = PB^2 + PD^2$.

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