Question

In the given figure if P, Q, R and S are the midpoints of side ab, ad, CD, BC respectively then 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

Answer 2

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