In $\Delta ABC$,the bisector of $\angle A$ intersects $\overline{BC}$ at $D$. If $AB = 8$,$AC = 10$ and $BD = 3.2$,then $DC = \ldots$

In $\Delta ABC$,$A-M-B$,$A-N-C$ and $\overline{MN} \parallel \overline{BC}$. If $\frac{AM}{MB} = \frac{4}{5}$ and $NC = 2.5$,find $AN$.

In $\Delta ABC$,$m\angle B = 90^{\circ}$ and $\overline{BD}$ is an altitude to the hypotenuse $\overline{AC}$. If $AC = 5 CD$,prove that $BD = 2 CD$.

$\Delta ABC \sim \Delta XYZ$ for the correspondence $ABC \leftrightarrow XYZ$. If $AB + BC = 7$,$XY + YZ = 10.5$,and $AC = 5$,then $XZ = \ldots$

Find the altitude of an equilateral triangle of side $8 \, cm$. (in $cm$)

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.