In $\Delta PQR$,$m \angle Q = 90^{\circ}$ and $\overline{QS}$ is an altitude. If $PS - SR = 10$ and $PQ^2 - QR^2 = 260$,then $PR = \dots$

In $\square ABCD$,$\overline{AB} \parallel \overline{CD}$ and $\overline{AC} \cap \overline{BD} = \{M\}$. If $MA = 3x - 19$,$MB = x - 3$,$MC = x - 5$,and $MD = 3$,find the value of $x$.

$\Delta ABC \sim \Delta DEF$ for the correspondence $ABC \leftrightarrow EFD$. $AB : BC : CA = 4 : 3 : 5$. If the perimeter of $\Delta DEF$ is $36$,find the measures of all the sides of $\Delta DEF$.

In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $\overline{BM}$ is an altitude to the hypotenuse $\overline{AC}$. If $AM = 4$ and $CM = 12$,find $AB$ and $BC$.

In a square,the length of a diagonal is $8$. Find the length of the sides of the square. (in $\sqrt{2}$)

$\Delta PQR \sim \Delta DEF$ for the correspondence $PQR \leftrightarrow DEF$. If $3 PQ = 2 DE$,$EF = 6$,and $PR = 8$,find $QR$ and $DF$.