Question

In figure, $ABCD$ is a parallelogram $P$ and $Q$ are the mid-points of opposite sides $AB$ and $DC$ of a parallelogram $ABCD$. Prove that $PRQS$ is a parallelogram.

Answer 1

In parallelogram $ABCD$,

$AB\parallel CD$

$\therefore AP\parallel CQ$ ...(1)

Similarly, $AB\cong CD$

$\therefore \frac{1}{2}AB\cong \frac{1}{2}CD$

$\therefore AP\cong CQ$ ...(2)

($\because P$ and $Q$ are mid-points of sides $AB$ and $CD$ respectively)

From (1) and (2), we can conclude that $APCQ$ is a parallelogram

Similarly, $PBQD$ is also a parallelogram.

Now, as $APCQ$ is a parallelogram,

$\therefore AQ\parallel PC$

$\therefore SQ\parallel PR$ ...(3)

Similarly, $PBQD$ is a parallelogram

$\therefore DP\parallel QB$

$\therefore SP\parallel QR$ ...(4)

From (3) and (4),

$PRQS$ is a parallelogram