In $\Delta PQR$,$m \angle Q = 90^\circ$. If $PQ : QR = 1 : 1$,then $PQ : PR = \ldots$

In the figure,if $\triangle ABC \sim \triangle DEF$ and their side lengths (in $cm$) are as marked,find the lengths of the sides of each triangle.

In $\square ABCD$,$T$ is a point on $\overline{BC}$. The line segment $\overrightarrow{AT}$ intersects $\overline{BD}$ at $M$ and the extension of $\overrightarrow{DC}$ at $O$. Prove that $AM^{2} = MT \times MO$.

In $\Delta ABC$,$m\angle B = 90^{\circ}$ and $\overline{BD}$ is an altitude to the hypotenuse $\overline{AC}$. If $AC = 5 CD$,prove that $BD = 2 CD$.

In $\Delta XYZ, m\angle Y = 90^{\circ}$. If $XY = YZ$ and $XZ = 10$,then $XY = \ldots$

In $\Delta ABC$,$\overline{AD}$ and $\overline{BE}$ are medians. Line $m$ through $D$ and parallel to $\overline{BE}$ intersects $\overline{AC}$ at $K$. If $EK = 2.5$,then $AC = \ldots$