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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

2
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is a square

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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

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