$\Delta PQR \sim \Delta XYZ$ for the correspondence $PQR \leftrightarrow XZY$. If $PQ = 8$,$XZ = 12$ and the area of $\Delta PQR$ is $24$,find the area of $\Delta XYZ$.

$A$ ladder is placed leaning on a wall. Its upper end reaches to the height of $12 \, m$ on the wall and its lower end rests $9 \, m$ away from the base of the wall. Find the length of the ladder in meters.

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 $\Delta ABC$,$\overline{AB} \cong \overline{AC}$ and $\overline{AM}$ is an altitude. If $AM = 15$ and the perimeter of $\Delta ABC$ is $50$,find the area of $\Delta ABC$.

In $\Delta PQR$,$PQ = 25$ and $QR = 24$. If $PR = \dots$,then $\angle R$ is a right angle.

In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $\overline{BD}$ is an altitude to the hypotenuse $\overline{AC}$. If $BD = 2 \sqrt{30}$ and $CD = 6$,then $AC = \ldots$