Q. 114.2( 6 Votes )

Show that the quadrilateral formed by joining the mid-points of the consecutive sides of a square is also a square.

Answer :


Let ABCD is a square.


AB = BC = CD = AD


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


Now, in ΔADC,


SR||AC and SR = AC


In ΔABC,


PQ||AC and PQ = AC


SR||PQ and SR = PQ = AC


Similarly,


SP||BD and BD||RQ


SP||RQ and SP = BD


And RQ = BD


SP = RQ = BD


Since, diagonals of a square bisect each other at right angles.


AC = BD


SP = RQ = AC


SR = PQ = SP = RQ


All sides are equal.


Now, in quad OERF,


OE||FR and OF||ER


EOF = ERF = 90


Hence, PQRS is a square.


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