In $\Delta ABC$,the bisector of $\angle A$ intersects $\overline{BC}$ at $D$. If $AB = 12$,$AC = 8$ and $BD = 9$,find $DC$.

In $\Delta ABC$,$\overline{AD}$ is a median. If $AB = 10$,$BC = 16$ and $AD = \sqrt{58}$,find $AC$.

In $\Delta ABC$,$\overline{AB} \cong \overline{AC}$ and $\overline{AD}$ is a median. If $BC = 12$ and $AD = 8$,find $AB$.

The perimeter of an equilateral triangle is $18$. Then,the length of its altitude is $\ldots \ldots \ldots$

$\Delta ABC \sim \Delta PQR$ for the correspondence $ABC \leftrightarrow PQR$. If $AB = 4, BC = 8, AC = 10$ and $PR = 15$,then the perimeter of $\Delta PQR = \dots$

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 $\overline{EF} \parallel \overline{BC}$.