Question

ABCD is a parallelogram in which P and Q are mid-points of opposite sides AB and CD . If AQ intersects DP at S and BQ intersects CP at R, show that ;
i) APCQ is a parallelogram
ii) DPBQ is a parallelogram
iii) PSQR is a parallelogram

Answer 1

Given:$ABCD$ is a parallelogram

$P,Q$ are the midpoints of $AB$ and $CD$ respectively.

$\left(i\right)$Since $AB\parallel DC$ since opposite sides of parallelogram are parallel.

$\Rightarrow AP\parallel QC$ since parts of parallel lines are parallel

Also,$AB=CD$ since opposite sides of parallelogram are equal.

$\Rightarrow \frac{1}{2}AB=\frac{1}{2}CD$

$\Rightarrow AP=QC$ since $P$ is mid-point of $AB$ and $Q$ is the midpoint of $DC$

Since $AP\parallel QC$ and $AP=AC$

One pair of opposite sides of $APCQ$ are equal and parallel

$\therefore APCQ$ is a parallelogram.

$\left(ii\right)$Since $AN\parallel DC$ since opposite sides of parallelogram are parallel.

$\Rightarrow PB\parallel DQ$ since parts of parallel lines are parallel

Also,$AB=CD$ since opposite sides of parallelogram are equal.

$\Rightarrow \frac{1}{2}AB=\frac{1}{2}CD$

$\Rightarrow PB=DQ$ since $P$ is mid-point of $AB$ and $Q$ is the midpoint of $DC$

Since $PB\parallel DQ$ and $PB=DQ$

One pair of opposite sides of $DPBQ$ are equal and parallel

$\therefore DPBQ$ is a parallelogram.

$\left(iii\right)$So,$DP\parallel QB$ since opposite sides of parallelogram are parallel.

$\Rightarrow SP\parallel QR$ since parts of parallel lines are parallel .......$\left(1\right)$

Also,in part$\left(i\right)$ we proved $APCQ$ is a parallelogram

So,$AQ\parallel PC$ since opposite sides of parallelogram are parallel

$\therefore SQ\parallel PR$ since parts of parallel lines are parallel ........$\left(2\right)$

From $\left(1\right)$ and $\left(2\right)$ we have $SP\parallel QR$ and $SQ\parallel PR$

Now, in $PSQR$ both pairs of opposite sides of $PSQR$ are paralle.

$\therefore PSQR$ is a parallelogram.