In $\Delta PQR$, $m \angle Q = 90^{\circ}$ and $\overline{QM}$ is an altitude to the hypotenuse $PR$. If $QM = 14$ and $RM = 7$, find $PQ$. (in $\sqrt{5}$)

In $\Delta ABC$,$m \angle B = 90^\circ$ and $\overline{BM}$ is an altitude to the hypotenuse $AC$. If $BC = 2x + 2$,$CM = x + 1$,and $AC = 5x$,find the value of $x$.

It is given that $\triangle FED \sim \triangle STU$. Is it true to say that $\frac{DE}{ST} = \frac{EF}{TU}$? Why?

$\Delta ABC \sim \Delta PQR$ for the correspondence $ABC \leftrightarrow PQR$. $\overline{AD}$ and $\overline{PM}$ are medians of these triangles. Prove that $AB \times PM = PQ \times AD$.

Sides of two similar triangles are in the ratio $4:9$. Areas of these triangles are in the ratio

In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $\overline{BD}$ is an altitude to the hypotenuse $AC$. If $AD = 9$ and $CD = 4$,find $BD$.