The perimeter of an equilateral triangle is $18$. Then,the length of its altitude is $\ldots \ldots \ldots$

The length of a diagonal of a square is $6$. Then,its area is............

In $\Delta PQR$,$\overline{QM}$ is an altitude. If $PQ = 2PM$ and $RM = 3PM$,prove that $\Delta PQR$ is a right-angled triangle.

In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $\overline{BM}$ is an altitude to the hypotenuse $\overline{AC}$. If $AM = 4$ and $CM = 12$,find $AB$ and $BC$.

If $\Delta ABC \sim \Delta PQR$ under the correspondence $ABC \leftrightarrow PQR$. If $AB = 12$,$\text{Area}(\Delta ABC) = 36$,and $\text{Area}(\Delta PQR) = 64$,then $PQ = \ldots$

From an airport,two planes begin their journey at the same instant to fly at the same height. One plane moves towards north at an average speed of $800 \, km/h$ and the other plane moves towards east at an average speed of $840 \, km/h$. Find the distance in $km$ between those planes after $1 \frac{1}{2}$ hours of journey.