In $\Delta PQR$,$m\angle Q = 90^{\circ}$ and $T$ is the midpoint of $\overline{PR}$. If $PQ = 6$ and $QR = 8$,then $QT = \ldots$

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 $\Delta ABC$,$A-M-B$,$A-N-C$ and $\overline{MN} \parallel \overline{BC}$. If $AM = x+3$,$AB = 2x$,$AN = x+5$ and $AC = 2x+3$,find the value of $x$.

In $\Delta ABC$,$m\angle B = 90^{\circ}$. If $AB : BC = 3 : 4$,find $AB : AC$.

In $\Delta XYZ$,$\overline{XN}$ is a median. If $XY^2 + XZ^2 = 200$ and $XN = 8$,find $YZ$.

$A$ 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.