In $\Delta ABC$,$m \angle B = 90^{\circ}$. If $AB = 4$ and $BC = 7.5$,find $AC$.

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 $\overline{BM}$ is an altitude to the hypotenuse $\overline{AC}$. If $AB = x - 2$,$AM = x - 6$,and $AC = 2x - 4$,find the value of $x$.

In $\Delta ABC$,$AB > AC$ and $D$ is the midpoint of $\overline{BC}$. $\overline{AM} \perp \overline{BC}$ and $M \in \overline{BC}$. Prove that $AB^{2} - AC^{2} = 2 \cdot BC \cdot DM$.

The vertices of a $\Delta ABC$ are $A(4, 6)$,$B(1, 5)$,and $C(7, 2)$. $A$ line is drawn to intersect sides $AB$ and $AC$ at $D$ and $E$ respectively,such that $\frac{AD}{AB} = \frac{AE}{AC} = \frac{1}{4}$. Calculate the area of $\Delta ADE$ and compare it with the area of $\Delta ABC$.

In $\Delta ABC$,if $\ldots \ldots \ldots \ldots$,then by Apollonius' theorem,$AB^{2} + AC^{2} = 2(AD^{2} + BD^{2})$ holds good.