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

Explanation

Solution
In a square `ABCD, P,Q,R` and `S` are the mid-points of `AB,BC,CD` and `DA`
respectively.
⇒
`AB=BC=CD=AD` [ Sides of square are equal ]
In`/_\ADC`
,
`SR∥AC and SR= 1/2 AC` [ By mid-point theorem ] ---- ( 1 )
In`/_\ABC`,
`PQ∥AC` and `PQ= 1/2 AC` [ By mid-point theorem ] ---- ( 2 )

From equation ( 1 ) and ( 2 ),

`SR∥PQ` and `SR=PQ= 1/2 AC` ---- ( 3 )
Similarly, `SP∥BD` and `BD∥RQ`
∴ `SP∥RQ` and `SP= 1/2 BD`
and `RQ=
1/2 BD`
∴ `SP=RQ=
1/2 BD`
Since, diagonals of a square bisect each other at right angle.
∴ `AC=BD`

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