In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $\overline{BM}$ is an altitude to the hypotenuse $\overline{AC}$. If $AM = 2x$,$BM = 3x + 5$,and $CM = 8x$,find the value of $x$.

$ABCD$ is a rectangle. If $AB = 7.5$ and $AC = 19.5$,find $BC$.

In $\Delta ABC$,$m \angle B = 90^{\circ}$ and $\overline{BD}$ is an altitude to the hypotenuse $AC$. If $AD = 9$ and $CD = 27$,find $AB$.

Using Theorem $6.1,$ prove that a line drawn through the mid-point of one side of a triangle parallel to another side bisects the third side.

$A$ line drawn parallel to $\overline{YZ}$ in the plane of $\Delta XYZ$ passes through the midpoint of $\overline{XY}$. Prove that this line bisects $\overline{XZ}$.

In $\Delta ABC$,$\overline{AD}$ is a median. The bisectors of $\angle ADB$ and $\angle ADC$ intersect $\overline{AB}$ and $\overline{AC}$ at $E$ and $F$ respectively. Prove that $\overline{EF} \parallel \overline{BC}$.