In $\Delta ABC$,$m\angle B = 90^{\circ}$ and $\overline{BM}$ is an altitude to the hypotenuse $\overline{AC}$. If $AM = 12$ and $CM = 3$,then $BM = \dots$

In $\Delta ABC$,$AB = 3$,$BC = 6$,and $AC = 8$. In $\Delta PQR$,$PQ = 13.6$,$QR = 10.2$,and $PR = 5.1$. Which correspondence between $\Delta ABC$ and $\Delta PQR$ is a similarity?

The perimeter of rhombus $ABCD$ is $68$. If $AC = 30$,find $BD$.

If in two triangles $ABC$ and $PQR$,$\frac{AB}{QR} = \frac{BC}{PR} = \frac{CA}{PQ}$,then

In $\Delta ABC$,$m\angle B = 90^{\circ}$ and points $D$ and $E$ trisect $\overline{BC}$. Prove that $8AE^{2} - 3AC^{2} = 5AB^{2}$. (Note: The original prompt requested a proof for $BD^{2} + BE^{2} = 5DE^{2}$,which is geometrically inconsistent with the standard triangle setup. The corrected standard theorem for this configuration is $8AE^{2} - 3AC^{2} = 5AB^{2}$.)

In $\Delta PQR$,$\overline{PM}$ is a median. If $PQ^{2} + PR^{2} = 148$ and $PM = 7$,find $QR$.