In $\Delta ABC$,$\overline{AD}$ is a median. If $AB = 8$,$AC = 15$ and $AD = 8.5$,find $BC$.

In $\square ABCD$,$P \in \overline{AB}$,$Q \in \overline{BC}$,$R \in \overline{CD}$,and $S \in \overline{DA}$ such that $\frac{AP}{PB} = \frac{AS}{SD}$ and $\frac{CB}{QB} = \frac{CR}{RD}$. Prove that $\overline{PS} \parallel \overline{QR}$.

In $\Delta ABC$,points $D$ and $E$ lie on $\overline{AB}$ such that $A-D-E-B$ and $AD = EB$. $P \in \overline{AC}$ and $Q \in \overline{BC}$ such that $\overline{DP} \parallel \overline{BC}$ and $\overline{EQ} \parallel \overline{AC}$. Prove that $\overline{PQ} \parallel \overline{AB}$.

In the figure,if $\angle ACB = \angle CDA$,$AC = 8 \, cm$ and $AD = 3 \, cm$,find $BD$ (in $cm$).

In $\Delta PQR$,$M$ and $N$ are the midpoints of $\overline{PQ}$ and $\overline{PR}$ respectively. If the area of $\Delta PMN$ is $24$,find the area of $\Delta PQR$.

Which of the following correctly matches the information in Part $I$ and Part $II$?

Part $I$	Part $II$
$1.$ In $\Delta ABC$ and $\Delta PQR, \angle A \cong \angle P$ and $\angle C \cong \angle Q$	$a.$ Correspondence $ABC \leftrightarrow RQP$ is a similarity.
$2.$ In $\Delta ABC$ and $\Delta PQR, \frac{AB}{QR} = \frac{BC}{PQ}$ and $\angle B \cong \angle Q$	$b.$ Correspondence $ABC \leftrightarrow QPR$ is a similarity.
$3.$ In $\Delta ABC$ and $\Delta PQR, \frac{AB}{PQ} = \frac{BC}{PR} = \frac{CA}{QR}$	$c.$ Correspondence $ABC \leftrightarrow PQR$ is a similarity.
$4.$ In $\Delta ABC$ and $\Delta PQR, \frac{AB}{PQ} = \frac{CA}{PR}$ and $\angle A \cong \angle P$	$d.$ Correspondence $ABC \leftrightarrow PRQ$ is a similarity.