Question

In the given figure if P, Q, R and S be the
mid-points of sides AB, BC, CD and DA
respectively, prove that PQRS is a
parallelogram​

Answer 1

Answer:

Given, P, Q, R and S are the mid-points of sides AB, AD, CD, and BC respectively.

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

Since, P is the mid point of AB

So, AP = PB                1

Since, Q is the mid point of BC

So, QC = QB               2

Since, R is the mid point of CD

So, CR = RD                 3

Since, S is the mid point of AD

So, AS = SD                    4  

Now, divide equation 1 by equation 4, we get

AP/AS = PB/SD

=> AP/PB = AS/SD

=> PS || BD                         5 {converse of Thales Theorem}

Similarly, QR || BD                6

Again, From equation 5 and 6, we get

PS || QR                                  7

Now, divide equation 1 by equation 2, we get

AP/QC = PB/QB

=> AP/PB = QC/QB

=> PQ || AC                               8 {converse of Thales Theorem}

Similarly, SR || AC                     9

From 8 and 9, PQ || SR   10

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

Hence, PQRS is a parallelogram

Explanation:

Answer 2

Answer:

PQ||SR and PQ=SR

Explanation:

Join AC. By Mid-Point Theorem, we have

PQ||AC and PQ=1/2 AC ..........(1)

and

SR||AC and SR=1/2AC. ...........(2)

From equation (1) and (2), we get

PQ||SR and PQ=SR

Therefore, PQRS is a parallelogram.