In $\Delta ABC$,$P$ and $Q$ are the midpoints of $AB$ and $AC$ respectively. If $PQ + BC = 8.4 \, cm$ and $AB + AC = 20.5 \, cm$,then find the perimeter of $\Delta ABC$ in $cm$.

$(1)$ The diagonals of a rhombus are ....... to each other.
$(2)$ The diagonals of a parallelogram ...... each other.

In a quadrilateral $ABCD$,$\angle A : \angle B : \angle C : \angle D = 4 : 5 : 4 : 5$. Find the measure of each angle of the quadrilateral and state the type of quadrilateral $ABCD$.

In $\Delta ABC$, $P$, $Q$, and $R$ are the midpoints of $AB$, $BC$, and $CA$ respectively. If $AB = 8.2 \text{ cm}$, $BC = 6.4 \text{ cm}$, and $CA = 7.5 \text{ cm}$, then find the perimeter of quadrilateral $PBQR$. (in $\text{ cm}$)

Diagonals $AC$ and $BD$ of a quadrilateral $ABCD$ intersect each other at $O$ such that $OA : OC = 3 : 2$. Is $ABCD$ a parallelogram? Why or why not?

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 $, cm$)