Q. 54.1( 9 Votes )

P, Q, R and S are

Answer :


Given, P, Q, R and S are mid-points of the sides AB, BC, CD and DA, respectively.


Also, AC = BD and AC is perpendicular to BD.


In ΔADC, by mid-point theorem,


SR||AC and SR = AC


In ΔABC, by mid-point theorem,


PQ||AC and PQ = AC


PO||SR and PQ = SR = AC


Now, in ΔABD, by mid-point theorem,


SP||BD and SP = BD = AC


In ΔBCD, by mid-point theorem,


RQ||BD and RQ = BD = AC


SP = RQ = AC


PQ = SR = SP = RQ


Thus, all four sides are equal.


Now, in quadrilateral EOFR,


OE||FR, OF||ER


EOF = ERF = 90(Opposite angles of parallelogram)


QRS = 90


Hence, PQRS is a square.


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