In $\Delta PQR$,$P-X-Q$,$P-Y-R$ and $\overline{XY} \parallel \overline{QR}$. If $PX = 3$,$PQ = 8$ and $PY = 4.2$,find $YR$.

$O$ is the point of intersection of the diagonals $AC$ and $BD$ of a trapezium $ABCD$ with $AB \parallel DC$. Through $O$,a line segment $PQ$ is drawn parallel to $AB$ meeting $AD$ in $P$ and $BC$ in $Q$. Prove that $PO = QO$.

In $\Delta ABC$,$\overline{AD}$ is a median. If $AB^2 + AC^2 = 338$ and $AD = 5$,find $BC$.

In $\Delta ABC,$ the bisector of $\angle A$ intersects $\overline{BC}$ at $D.$ If $AB = 6,$ $BD = 4$ and $DC = 3,$ find $AC$.

In a rectangle $ABCD$,if $AB^{2} + BC^{2} = 64$,then find the length of the diagonal $AC$.

In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $\overline{BD}$ is an altitude to the hypotenuse $AC$. If $AD = 9$ and $CD = 27$,find $AB$.