Question

P,Q,R,S are mid points of the sides AB,BC,CD,DA of a quadrilateral.in which AC=BD,AC perpendicular BD.prove that PQRS
is a square

Answer 1

SP=DB BY MIDPOINT THEOREM -1
QR=DB BY MIDPOINT THEOREM -2
FROM 1 & 2 WE GET THAT SP=QR-3
HENCE IN THE SAME WAY SR=PQ-4
FROM 3 AND 4 WE CAN SAY THAT IT IS A PARLLELOGRAM
HENCE PQRS IS A PARLLELOGRAM
AS AC =BD GIVEN
HENCE SR=QR=QP=PS
HENCE PQRS IS A PARLLELOGRAM IN WHICH ALL SIDES ARE EQUAL
HENCE WE CAN SAY THAT PQRS IS A SQUARE
(THIS QUESTION IS FULLY BASES ON MIDPOINT THEOREM)
HOPE SO IT IS CORRCT

Answer 2

PQ is parallel to AC
SR is parallel to AC
NOW,PQRS is a parallelogram
T.P it is a square
in triangle PBQ and QRC
BQ=QC &RC=BP
angle B=C
BY SAS congruency
triangle CQR = QBP
BY corresponding parts of congruent triangles
PQ=OR
IN A SQUARE ADAJACENT SIDES ARE EQUAL