$ABCD$ is a quadrilateral in which $P$,$Q$,$R$ and $S$ are mid-points of the sides $AB$,$BC$,$CD$ and $DA$ (see Fig). $AC$ is a diagonal. Show that: $PQRS$ is a parallelogram.
(N/A) To prove that $PQRS$ is a parallelogram.
In $\Delta ABC$,$P$ and $Q$ are the mid-points of $AB$ and $BC$.
$\therefore PQ = \frac{1}{2} AC$ and $PQ \parallel AC$ .......... $(1)$
In $\Delta ACD$,$S$ and $R$ are the mid-points of $DA$ and $CD$.
$\therefore SR = \frac{1}{2} AC$ and $SR \parallel AC$ .......... $(2)$
From $(1)$ and $(2)$,we get
$PQ = \frac{1}{2} AC = SR$ and $PQ \parallel AC \parallel SR$
$\Rightarrow PQ = SR$ and $PQ \parallel SR$
i.e.,one pair of opposite sides in quadrilateral $PQRS$ is equal and parallel.
$\therefore PQRS$ is a parallelogram.
In $\Delta ABC$,$P$ and $Q$ are the mid-points of $AB$ and $BC$.
$\therefore PQ = \frac{1}{2} AC$ and $PQ \parallel AC$ .......... $(1)$
In $\Delta ACD$,$S$ and $R$ are the mid-points of $DA$ and $CD$.
$\therefore SR = \frac{1}{2} AC$ and $SR \parallel AC$ .......... $(2)$
From $(1)$ and $(2)$,we get
$PQ = \frac{1}{2} AC = SR$ and $PQ \parallel AC \parallel SR$
$\Rightarrow PQ = SR$ and $PQ \parallel SR$
i.e.,one pair of opposite sides in quadrilateral $PQRS$ is equal and parallel.
$\therefore PQRS$ is a parallelogram.
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