In $\triangle ABC$,$AB = 5\, cm$,$BC = 8\, cm$,and $CA = 7\, cm$. If $D$ and $E$ are respectively the mid-points of $AB$ and $BC$,determine the length of $DE$. (in $, cm$)

$ABCD$ is a parallelogram and $P$ and $Q$ are points on the diagonal $AC$ such that $AP = PQ = QC$. Prove that $BQ \parallel DP$ and $BD$ bisects $PQ$.

If $APB$ and $CQD$ are two parallel lines,then the bisectors of the angles $APQ, BPQ, CQP$ and $PQD$ form

In $\Delta ABC$,$P$,$Q$,and $R$ are the midpoints of $AB$,$BC$,and $CA$ respectively. If $AB = 8 \text{ cm}$,$BC = 6.6 \text{ cm}$,and $CA = 5.4 \text{ cm}$,then find the perimeter of $\Delta PQR$ in $\text{cm}$. (in $.0$)

In quadrilateral $ABCD$,$AC = BD$ and $AC \perp BD$. If $P, Q, R$ and $S$ are the midpoints of the sides $AB, BC, CD$ and $DA$ respectively,prove that $PQRS$ is a square.

State whether each of the following statements is true or false:
$(1)$ $ABCD$ is a rhombus. If $AB = 12 \, \text{cm}$, then the perimeter of $ABCD$ is $48 \, \text{cm}$.
$(2)$ $PQRS$ is a square. If $PR = 8 \, \text{cm}$, then the perimeter of $PQRS$ is $32 \, \text{cm}$.