The perimeter of square $ABCD$ is $48$. Find the length of its diagonal $\overline{AC}$. (in $\sqrt{2}$)

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 $\Delta ABC$,$m \angle A = 90^{\circ}$. If $AB = 3x - 2$,$AC = 5x + 4$,and $BC = 6x + 2$,find the value of $x$.

In $\Delta PQR$,$m \angle P = 90^{\circ}$ and $\overline{PM}$ is an altitude. If $QM = 2x^{2}$ and $RM = 8x^{2}$,then $PM = \ldots \ldots$

$A$ and $B$ are points on the sides $PQ$ and $PR$ of a triangle $PQR$ respectively,such that $PQ = 12.5 \, cm$,$PA = 5 \, cm$,$BR = 6 \, cm$,and $PB = 4 \, cm$. Is $AB \parallel QR$? Give reasons for your answer.

In $\Delta XYZ$,$m\angle Y = 90^{\circ}$ and $\overline{YM}$ is an altitude to the hypotenuse $\overline{XZ}$. If $XM = \sqrt{12}$ and $ZM = \sqrt{3}$,find $YM$.