The length of a diagonal of a square is $8 \sqrt{2}$. Then,the perimeter of the square is..............

If in two triangles $ABC$ and $PQR$,$\frac{AB}{QR} = \frac{BC}{PR} = \frac{CA}{PQ}$,then

In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $\overline{BM}$ is an altitude. If $AM - CM = 10$ and $AB^{2} - BC^{2} = 260$,find $AC$.

$\Delta PQR \sim \Delta XYZ$ for the correspondence $PQR \leftrightarrow XYZ$. The perimeter of $\Delta PQR$ is $24$ and the perimeter of $\Delta XYZ$ is $60$. If $PR = 10$,find $XZ$.

If $\Delta ABC \sim \Delta XYZ$ under the correspondence $ABC \leftrightarrow XZY$,and the perimeter of $\Delta ABC$ is $45$,the perimeter of $\Delta XYZ$ is $30$,and $AB = 21$,find the length of $XZ$.

In $\triangle PQR$,$PD \perp QR$ such that $D$ lies on $QR$. If $PQ = a$,$PR = b$,$QD = c$,and $DR = d$,prove that $(a+b)(a-b) = (c+d)(c-d)$.