In $\Delta PQR$,$m\angle Q = 90^{\circ}$ and $\overline{QM}$ is an altitude. If $PM = x$ and $RM = y$,find the lengths of $\overline{PQ}$,$\overline{QR}$,$\overline{PR}$,and $\overline{QM}$ in terms of $x$ and $y$.

(N/A) In $\Delta PQR$,$m\angle Q = 90^{\circ}$ and $\overline{QM}$ is an altitude to the hypotenuse $\overline{PR}$.
Since $M$ lies on $\overline{PR}$,we have $PR = PM + RM = x + y$.
Using the geometric mean theorem for the altitude to the hypotenuse:
$QM^2 = PM \cdot RM = x \cdot y$
$\therefore QM = \sqrt{xy}$.
Using the leg rule for $\Delta PQR$:
$PQ^2 = PM \cdot PR = x(x + y) = x^2 + xy$
$\therefore PQ = \sqrt{x^2 + xy}$.
Similarly,for the other leg:
$QR^2 = RM \cdot PR = y(x + y) = y^2 + xy$
$\therefore QR = \sqrt{y^2 + xy}$.
Thus,the lengths are $PQ = \sqrt{x^2 + xy}$,$QR = \sqrt{y^2 + xy}$,$PR = x + y$,and $QM = \sqrt{xy}$.