In $\Delta ABC$,$m \angle A = 90^{\circ}$ and $\overline{AD}$ is an altitude to the hypotenuse $BC$. Then,$AD^{2} = \ldots$

In $\Delta ABC$ and $\Delta XYZ$,$m \angle A = m \angle X$ and $m \angle B = m \angle Y$. $\overline{AM}$ is a median of $\Delta ABC$ and $\overline{XP}$ is a median of $\Delta XYZ$. Prove that $AM \times YZ = XP \times BC$.

In $\Delta ABC$,$AB + BC = 23$,$BC + AC = 32$,and $AB + AC = 25$. Show that $\Delta ABC$ is a right-angled triangle.

In $\Delta ABC,$ $\overline{AD}$ is a median. If $AD=7,$ $AC=7$ and $BD=4,$ find the perimeter of $\Delta ABC.$

In $\Delta ABC$,$A-P-B$,$A-Q-C$ and $\overline{PQ} \parallel \overline{BC}$. If $AP = 8x-7$,$PB = 5x-3$,$AQ = 4x-3$ and $QC = 3x-1$,find the value of $x$.

In $\Delta XYZ$ and $\Delta MNO$,$\frac{XY}{MN} = \frac{XZ}{NO} = \frac{YZ}{MO}$. Then,the correspondence $\ldots$ is a similarity.