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

$ABCD$ is a trapezium in which $AB \parallel DC$ and $P$ and $Q$ are points on $AD$ and $BC$ respectively such that $PQ \parallel DC$. If $PD = 18 \, cm$,$BQ = 35 \, cm$ and $QC = 15 \, cm$,find $AD$ (in $cm$).

$A$ $8 \, m$ long bamboo tree standing erect on the ground breaks at the height of $3 \, m$ from the ground. The broken part of the tree remains attached to the trunk. Find the distance in $m$ between the top of the tree on the ground and the base of the tree.

In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $\overline{BM}$ is an altitude to the hypotenuse $\overline{AC}$. If $AB = x - 2$,$AM = x - 6$,and $AC = 2x - 4$,find the value of $x$.

In $\Delta PQR$,$m\angle Q = 90^\circ$. If $PQ = 8$ and $QR = 15$,then $PR = \ldots$

In $\Delta ABC$,$\overline{AB} \cong \overline{AC}$ and $\overline{AM}$ is an altitude. If $AM = 15$ and the perimeter of $\Delta ABC$ is $50$,find the area of $\Delta ABC$.