Question

ABCD is a quadrilateral in which P,Q,R and S are mid-point of the sides AB,BC,CD and DA respectively.BD is a diagonal. Show that PQRS is a parallelogram​

Answer 1

$\sf\huge\bold{Answer:}$

Given :

ABCD is a quadrilateral.

P is the mid point of AB

Q is the mid point of BC

R is the mid point of CD

S is the mid point of DA

BD is a diagonal

To show :

PQRS is a parallelogram

i.e RQ||PS

QR = PS

Solution :

In △DBC 

Q is the mid point of BC and

R is the mid point of DC.

Therefore,  QR || BD ... (I)

QR=½ BD...(II)

[By mid-point theorem.]

In △BAD

P is the mid point of AB and

S is the mid point of AD.

Therefore,  PS|| BD...(III)

 PS=½BD...(IV)

[By mid-point theorem.]

What is Mid point Theorem?

[ The Midpoint Theorem states that the segment joining two sides of a triangle at the midpoints of those sides is parallel to the third side and is half the length of the third side.]

$\fbox{1.}$ To show QR || PS

From I and III

QR||BD||PS

Thus, QR||PS...(i)

$\fbox{2.}$ To show QR=PS

From II and IV

QR=½BD=PS

Thus, PQ=SR...(ii)

Hence, From (i) and (ii) it is proved that

$\sf\pink{PQRS \:is\: a\: parallelogram }$