The length of a diagonal of a square is $5 \sqrt{2}$. Then,the length of its sides is............

In $\Delta ABC$,$m \angle B = 90$ and $D$ is the midpoint of $\overline{AC}$. Then,$BD = \dots$

$\Delta ABC \sim \Delta PQR$ for the correspondence $ABC \leftrightarrow PQR$. If $AB : PQ = 3 : 4$ and the perimeter of $\Delta PQR$ is $24$,find the perimeter of $\Delta ABC$.

In triangles $PQR$ and $MST$,$\angle P = 55^{\circ}$,$\angle Q = 25^{\circ}$,$\angle M = 100^{\circ}$ and $\angle S = 25^{\circ}$. Is $\triangle QPR \sim \triangle TSM$? Why?

In $\square ABCD$,$AB = AD$. The bisector of $\angle BAC$ intersects $\overline{BC}$ at $E$ and the bisector of $\angle DAC$ intersects $\overline{DC}$ at $F$. Prove that $\overline{EF} \parallel \overline{BD}$.

In $\Delta ABC$,$AB = 3$,$BC = 6$,and $AC = 8$. In $\Delta PQR$,$PQ = 13.6$,$QR = 10.2$,and $PR = 5.1$. Which correspondence between $\Delta ABC$ and $\Delta PQR$ is a similarity?