In $\Delta PQR$,$M$ and $N$ are the midpoints of $\overline{PQ}$ and $\overline{PR}$ respectively. If the area of $\Delta PMN$ is $24$,find the area of $\Delta PQR$.

In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $\overline{BM}$ is an altitude to the hypotenuse $\overline{AC}$. If $AM = 4$ and $CM = 5$,find $AB$ and $BM$.

In quadrilateral $ABCD$,$P$ is the midpoint of $\overline{BC}$. If $\overline{DP}$ produced intersects $\overline{AB}$ produced at $Q$,prove that $AB = 2CD$ given that $ABCD$ is a parallelogram.

In $\Delta ABC$,$m \angle B = 90^{\circ}$,$AC - BC = 2$ and $BC - AB = 7$. Find the perimeter of $\Delta ABC$.

Given $\Delta ABC \sim \Delta PQR$ for the correspondence $ABC \leftrightarrow PQR$. If $2 m \angle P = 3 m \angle Q$ and $m \angle C = 100^{\circ}$,find $m \angle B$. (in $^{\circ}$)

The perimeter of equilateral $\Delta ABC$ is $24$. Find the length of its altitude. (in $\sqrt{3}$)