In $\Delta ABC$,$\overline{AD}$ is a median. If $AB = 8$,$AC = 15$ and $AD = 8.5$,find $BC$.

In $\Delta PQR,$ the midpoints of the sides are $X, Y, Z.$ In $\Delta XYZ,$ the midpoints of the sides are $A, B, C.$ If the area of $\Delta PQR$ is $240,$ find the area of $\Delta XYZ$ and $\Delta ABC.$

In $\Delta ABC$,$A-M-B$,$A-N-C$ and $\overline{MN} \parallel \overline{BC}$. If $AM = 2$,$MB = 3$ and $MN = 3$,find $BC$.

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

For going to a city $B$ from city $A$,there is a route via city $C$ such that $AC \perp CB$,$AC = 2x \, km$ and $CB = 2(x+7) \, km$. It is proposed to construct a $26 \, km$ highway which directly connects the two cities $A$ and $B$. Find how much distance will be saved in reaching city $B$ from city $A$ after the construction of the highway (in $km$).

In $\Delta ABC$, $M$ and $N$ are the midpoints of $\overline{AB}$ and $\overline{AC}$ respectively. If the area of $\Delta ABC$ is $90$, find the area of $\Delta AMN$.