In $\Delta XYZ$,$m\angle Y = 90^{\circ}$ and $\overline{YM}$ is an altitude to the hypotenuse $\overline{XZ}$. If $XM = \sqrt{12}$ and $ZM = \sqrt{3}$,find $YM$.

In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $\overline{AB} \cong \overline{BC}$. Find the ratio $AB : AC$.

In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $\overline{BE}$ is a median. If $AB = 3.6$ and $BC = 4.8$,find $BE$.

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: $PQ = 8, QR = 6, PR = 12$.

In $\Delta ABC$,$\overline{AD}$ and $\overline{BE}$ are medians. Line $m$ through $D$ and parallel to $\overline{BE}$ intersects $\overline{AC}$ at $K$. If $EK = 2.5$,then $AC = \ldots$

In $\Delta ABC$,$m \angle A = 90^{\circ}$. If $AB = 3x - 2$,$AC = 5x + 4$,and $BC = 6x + 2$,find the value of $x$.