Prove that the perimeter of a midsegment triangle is half the perimeter of the triangle.

In the figure given below, $S$ and $T$ are the midpoints of $PQ$ and $PR$ respectively. If $ST=3 \mathrm{cm}$ then $QR=$ _____ $\mathrm{cm}$.

In the figure given below, $\mathrm{D}$ and $\mathrm{E}$ are the midpoints of $\mathrm{AB}$ and $\mathrm{AC}$ respectively. If $\mathrm{DE}=7.5 \mathrm{cm}$, then $\mathrm{BC}=$_____ $\mathrm{cm}$.

In $\Delta ABC$, $D$ is the midpoint of $AB$, $E$ is the midpoint of $AC$. If $DE=l \mathrm{cm}$ when $BC=6 \mathrm{cm}$, find the value of $l$.

In the figure given below, $ABC$ is an isosceles $△$ with $AB=AC$ and $CP\parallel BA$ and $AP$ is the bisector of ext. $\angle CAD$ of $△ABC$. Prove that
$\angle PAC=\angle BCA$. 

In the given Fig. $9.23, ABC$ is an isosceles $\Delta$ with $AB=AC$ and $CP\parallel BA$ and $AP$ is the bisector of ext. $\angle CAD$ of $∆ABC.$ Prove that
$ABCP$ is a parallelogram.