In $\Delta ABC$,$m \angle B = 90^{\circ}$. If $a = 9$ and $c = 12$,then $b = \dots$

In $\Delta ABC$,$m\angle B = 90^{\circ}$,$N \in \overline{AB}$ and $M \in \overline{BC}$. Prove that $AM^{2} + CN^{2} = AC^{2} + MN^{2}$.

In $\Delta ABC$,$m \angle B = 90^{\circ}$,$\overline{BM}$ is an altitude to the hypotenuse $\overline{AC}$,and $AM > CM$. If $BM = 12$ and $AC = 25$,find $AB$ and $BC$.

In $\Delta ABC$,$A-M-B$,$A-N-C$ and $\overline{MN} \parallel \overline{BC}$. If $AM = 3x - 1$,$MB = 2x + 1$,$AN = 3x + 1$ and $NC = 5x + 1$,find the value of $x$.

The lengths of the sides of a triangle are $m^{2}+n^{2}$,$2mn$,and $m^{2}-n^{2}$,where $m > n > 0$. Prove that the triangle is a right-angled triangle.

In the figure,$AD$ is a median of a triangle $ABC$ and $AM \perp BC$. Prove that $AC^{2} = AD^{2} + BC \cdot DM + \left(\frac{BC}{2}\right)^{2}$.