$A$ ladder $10\, m$ long reaches a window $8\, m$ above the ground. Find the distance of the foot of the ladder from the base of the wall. (in $, m$)

In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $\overline{BM}$ is an altitude to the hypotenuse $AC$. If $AB = 7$ and $BC = 24$,find the length of $BM$.

Below are given the measures of sides $\overline{PQ}$,$\overline{QR}$ and $\overline{PR}$ of $\Delta PQR$. In each case,determine whether $\Delta PQR$ is a right-angled triangle or not. If it is a right-angled triangle,state which angle is a right angle: $PQ = 7, QR = 24, PR = 25$.

In $\Delta ABC$,$m \angle B = 90^\circ$ and $\overline{BM}$ is an altitude to the hypotenuse $AC$. If $AM = BM = 8$,then $AC = \ldots$

In $\Delta PQR$,$m \angle Q = 90^{\circ}$ and $\overline{QS}$ is an altitude to the hypotenuse $PR$. If $PQ = 6$ and $PS = 4$,find $QS$,$QR$,and $RS$.

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 $YR$.