Question

In the given figure if P, Q, R and S be themid-points of sides AB, BC, CD and DArespectively, prove that PQRS is aparallelogram.
​

Answer 1

Step-by-step explanation:

Now, join AC, BD, PS, QR, PQ and RS

Since, P is the mid point of AB

So, AP = PB ......... ( i )

Since, Q is the mid point of BC

So, QC = QB ......... ( ii )

Since, R is the mid point of CD

So, CR = RD ...........( iii )

Since, S is the mid point of AD

So, AS = SD ..........( iv )

Now, divide equation 1 by equation 4, we get

AP/AS = PB/SD

=> AP/PB = AS/SD

=> PS || BD .........( v ) {converse of Thales Theorem}

Similarly, QR || BD ......... ( vi )

Again, From equation ( v ) and ( vi ), we get

PS || QR ( vii )

Now, divide equation ( i ) by equation ( ii ) we get

AP/QC = PB/QB

=> AP/PB = QC/QB

=> PQ || AC ........ ( viii ) {converse of Thales Theorem}

Similarly, SR || AC ......... ( ix )

From 8 and 9, PQ || SR .......... ( x )

From 7 and 10, PS | QR and PQ || SR

Hence, PQRS is a parallelogram ( Verified ).

MARK ME AS BRAINLIST ANSWER !!