In $\Delta PQR, m\angle Q = 90^{\circ}$ and $\overline{QM}$ is a median. If $PQ = 20$ and $QR = 21$,find $QM$.

In $\Delta ABC$,$m\angle B = 90^{\circ}$ and $\overline{BD}$ is an altitude to the hypotenuse $\overline{AC}$. If $AC = 5 CD$,prove that $BD = 2 CD$.

$\Delta XYZ \sim \Delta DEF$ for the correspondence $XYZ \leftrightarrow EDF$. If $XY = 3, YZ = 4, ZX = 6$ and $DF = 12$,find the perimeter of $\Delta DEF$.

In $\Delta ABC$,the bisector of $\angle A$ intersects $\overline{BC}$ at $D$. Prove that $BD = \frac{BC \times AB}{AB + AC}$ and $DC = \frac{BC \times AC}{AB + AC}$.

In $\Delta ABC$,$\Delta PQR$,and $\Delta XYZ$,the correspondences $ABC \leftrightarrow QPR$ and $PQR \leftrightarrow YZX$ are similarities. Then,the correspondence $ABC \leftrightarrow \dots$ is a similarity.

In parallelogram $ABCD$,$AB^{2} + BC^{2} = 260$ and $AC = 18$. Then $BD = \ldots$