If $\Delta PQR \sim \Delta XYZ$ for the correspondence $PQR \leftrightarrow ZYX$,and given $PQ = 6$,$QR = 8$,and $XY = 12$,find the length of $YZ$.

In the figure,$D$ is a point on the hypotenuse $AC$ of $\Delta ABC,$ such that $BD \perp AC,$ $DM \perp BC,$ and $DN \perp AB.$ Prove that $DM^{2} = DN \cdot MC.$

$\Delta PQR \sim \Delta XYZ$ for the correspondence $PQR \leftrightarrow XYZ$. If $PQ : QR : PR = 3 : 5 : 7$ and the perimeter of $\Delta XYZ$ is $22.5$,then $YZ = \ldots$

$\Delta PQR \sim \Delta XYZ$ for the correspondence $PQR \leftrightarrow XYZ$. The perimeter of $\Delta PQR$ is $24$ and the perimeter of $\Delta XYZ$ is $60$. If $PR = 10$,find $XZ$.

In the figure,two chords $AB$ and $CD$ of a circle are produced to intersect each other at point $P$ outside the circle. Prove that $PA \cdot PB = PC \cdot PD$.

In $\Delta ABC$, $P$ and $Q$ are the midpoints of $\overline{AB}$ and $\overline{AC}$ respectively. If the area of $\Delta APQ = 12 \sqrt{3}$, then the area of $\Delta ABC = \ldots$ (in $\sqrt{3}$)