Question

P,Q,R,S a mid points of sides
AB,BC,CD,DA of quadrilateral
ABCD & PR & QS intersect at E.
Prove PR & QS bisect at E​

Answer 1

Answer:

P,Q,R and S are the mid-point of the sides AB,BC,CD and DA of a quadrilateral ABCD.

⇒  AC=BD

In △ABC,

P and Q are the mid-points of the sides AB and BC respectively.

∴   PQ∥AC                      ----- ( 1 )

And PQ=  

2

1

​  

×AC                  ------ ( 2 )

Similarly, SR∥AC and  SR=  

2

1

​  

×AC         ----- ( 3 )

From ( 1 ), ( 2 ) and ( 3 ) we get,

 

⇒  PQ∥SR and PQ=SR=  

2

1

​  

×AC       ----- ( 4 )

Similarly we an show that,

⇒  SP∥RQ and SP=RQ=  

2

1

​  

×BD    ----- ( 5 )

Since, AC=BD

∴  PQ=SR=SP=RQ           [ From ( 4 ) and ( 5 ) ]

All sides of the quadrilateral are equal.

∴  PQRS is a rhombus.  

Step-by-step explanation:

Answer 2

P, Q, R and S are respectively the mid-points of the sides AB, BC, CD and DA of a quadrilateral ABCD in which AC = BD. Proving that PQRS is a rhombus

Firstly, use the mid-point theorem in various triangles of a quadrilateral. Further show that the line segments formed by joining the mid-points are equal, which prove the required quadrilateral.