Question

p,q,r,s are mid points of sides AB, BC , CD and DA respectively of a quardrilateral ABCD. Prove that ABCD is a parallelgram

Answer 1

Alliy, your question is wrong. You shall think to prove that PQRS is a parallelogram, shan't you? 

Ok, I'm proving PQRS is a parallelogram. 

After drawing quadrilateral ABCD and PQRS, draw diagonals AC, BD. 

Considering triangle ABC,
P & Q are midpoints. 
In any triangle, the line joining the midpoints of two sides is parallel to the third side, and its length is half of that of the third side also. 
Therefore, PQ || AC --- (1). 
And PQ = 1/2 AC --- (2). 

Considering triangle BCD, 
Q & R are midpoints. 
Therefore, QR || BD --- (3). 
And QR = 1/2 BD --- (4). 

Considering triangle CDA, 
R & S are midpoints. 
Therefore, RS || AC --- (5). 
And RS = 1/2 AC --- (6). 

Considering triangle DAB, 
P & S are midpoints. 
Therefore, PS || BD --- (7). 
And PS = 1/2 BD --- (8). 

From (1) & (5), we get, 
PQ || RS. 
And from (3) & (7), we get, 
QR || PS. 

Hence, opposite sides of quadrilateral PQRS are parallel. 

From (2) & (6), we get, 
PQ = RS. 
And from (4) & (8), we get, 
QR = PS. 

Hence, opposite sides of quadrilateral PQRS are equal also. 

Therefore, PQRS is a parallelogram. 

Hope this answer will be helpful.