In $\Delta ABC$,the bisector of $\angle B$ intersects $\overline{AC}$ at $D$. If $\frac{AD}{DC} = \frac{3}{4}$ and $AB = 7.5$,then $BC = \ldots$

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 $\square ABCD$,$P \in \overline{AB}$,$Q \in \overline{BC}$,$R \in \overline{CD}$,and $S \in \overline{AD}$ such that $\frac{AP}{AB} = \frac{CQ}{BC} = \frac{CR}{CD} = \frac{AS}{AD} = \frac{1}{3}$. Prove that $\square PQRS$ is a parallelogram.

In $\Delta ABC$,the bisectors of $\angle B$ and $\angle C$ intersect at $O$. $\overrightarrow{AO}$ intersects $\overline{BC}$ at $P$. Prove that $\frac{AB}{AC} = \frac{BP}{PC}$.

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

Using Theorem $6.2$,prove that the line joining the mid-points of any two sides of a triangle is parallel to the third side.