The perimeter of equilateral $\Delta ABC$ is $30$. Find its area. (in $\sqrt{3}$)

In $\Delta ABC$ and $\Delta XYZ$,$m \angle A = m \angle X$ and $m \angle B = m \angle Y$. $\overline{AM}$ is a median of $\Delta ABC$ and $\overline{XP}$ is a median of $\Delta XYZ$. Prove that $AM \times YZ = XP \times BC$.

In $\Delta ABC$ and $\Delta DEF$,$\angle A \cong \angle E$ and $m \angle A + m \angle B = m \angle D + m \angle E$. Then the correspondence $ABC \leftrightarrow \ldots$ between them is a similarity.

$\Delta DEF \sim \Delta PQR$ for the correspondence $DEF \leftrightarrow QPR$. If $2 DE = 3 PQ$ and $QR = 8$,then $DF = \ldots$

In $\Delta PQR$,$\overline{PM}$ is an altitude. If $PM^2 = QM \times RM$,prove that $\Delta PQR$ is a right-angled triangle.

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 $XZ$.