In $\Delta ABC$ and $\Delta PQR$,$m \angle A = 30^{\circ}$,$m \angle C = 60^{\circ}$,$m \angle P = 90^{\circ}$ and $m \angle Q = 30^{\circ}$. Then,the correspondence $ABC \leftrightarrow \ldots$ is a similarity.

Point $O$ lies in the interior of $\Delta ABC$. The bisectors of $\angle AOB, \angle BOC$ and $\angle COA$ intersect $\overline{AB}, \overline{BC}$ and $\overline{CA}$ at $D, E$ and $F$ respectively. Prove that $AD \times BE \times CF = DB \times EC \times FA$.

In $\Delta ABC$,$\overline{AD}$ is a median. If $AB = 8$ and $AD = BD = 8.5$,find $AC$.

Two adjacent sides of a rectangle measure $12$ and $35$. Then,the length of a diagonal of the rectangle is..........

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

In $\Delta ABC$ and $\Delta XYZ$,$\angle A \cong \angle Y$ and $\angle B \cong \angle Z$. If $\frac{AC}{YX} = \frac{5}{7}$ and $AB = 7$,find $YZ$.