Question

ABCD is a Parallelogram is which P and Q are mid-points of apposite 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
[3 MARKS]

Answer 1

Proof : 1 Mark each

(i) Since $ABCD$ is a parallelogram, $AB=DC$ and $AB||DC.$

$AP=\frac{1}{2}ABandQC=\frac{1}{2}DC$

Thus, $AP=QC$ and $AP||QC.$ Hence, $APCQ$ is a parallelogram.

$AQ||PC$ ------(a) (opposite sides of a parallelogram)

(ii) $BP=\frac{1}{2}ABandQD=\frac{1}{2}DC$

Thus, $BP=QD$ and $BP||QD$. Hence, $DPBQ$ is a parallelogram

$DP||BQ$ ---------(b) (opposite sides of a parallelogram)

(iii) From (a) and (b) we have

$AQ||PC\Rightarrow SQ||PR$

and $DP||BQ\Rightarrow SP||QR$

$\Rightarrow PSQR$ is a parallelogram.