Show that the quadrilateral formed by joining the mid-points of the consecutive sides of a square is also a square.
Hint: Prove all the sides are equal and diagonals are equal
Solution.
Given: In a square ABCD; P, Q, R and S are the mid-points of AB, BC, CD and DA.
To Prove: PQRS is a square
Construction: Join AC and BD
Proof : Here ABCD is a square
\ AB = BC = CD = AD
P, Q, R, S are the mid-points of AB, BC, CD and DA.
In ,
{using mid-point theorem} …..(1)
In ,
{using mid-point theorem} …..(2)
From equation 1 and 2
and …..(3)
Similarly, and
\ and {using mid-point theorem}
Since diagonals of a square bisect each other at right angles.
AC = BD
…..(4)
From equation 3 and 4
SP = PQ = SP = RQ {All sides are equal}
In quadrilateral OERF
and
In quadrilateral RSPQ
{Opposite angles in a parallelogram are equal}
Since all the sides are equal and angles are also equal, so PQRS is a square.
Hence Proved