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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