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

$A$ line drawn parallel to $\overline{YZ}$ in the plane of $\Delta XYZ$ passes through the midpoint of $\overline{XY}$. Prove that this line bisects $\overline{XZ}$.

In $\Delta ABC$,$A-M-B$,$A-N-C$ and $\overline{MN} \parallel \overline{BC}$. If $AM = x$,$MB = x-2$,$AN = x+2$ and $NC = x-1$,find the value of $x$.

In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $\overline{BM}$ is an altitude to the hypotenuse $\overline{AC}$. If $AM = 12$ and $CM = 3$,find $BM$.

If $\Delta ABC \sim \Delta MNP$ for the correspondence $ABC \leftrightarrow MNP,$ the perimeter of $\Delta ABC$ is $18$ and the perimeter of $\Delta MNP$ is $27.$ If $AB = 8,$ then $MN = \dots$

In $\Delta PQR$,$m\angle Q = 90^{\circ}$ and $T$ is the midpoint of $\overline{PR}$. If $PQ = 6$ and $QR = 8$,then $QT = \ldots$