Question

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

Answer 1

Given: In a square ABCD, P, Q, R and S are the mid-points of AB, BC, CD and DA, respectively.

To show: PQRS is a square.

Construction: Join AC and BD.

Proof: Since, ABCD is a square.

∴ AB = BC = CD = AD

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

Then, in ΔADC, SR || AC

And SR = `1/2`AC   [By mid-point theorem] ...(i)

In ΔABC, PQ || AC

And PQ = `1/2`AC  ...(ii)

From equations (i) and (ii),

SR || PQ and SR = PQ = `1/2`AC  ...(iii)

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

⇒ SP = RQ = `1/2`AC   ...(iv)

From equations (iii) and (iv),

SR = PQ = SP = RQ  ...[All side are equal]

Now, in quadrilateral OERF,

OE || FR and OF || ER

∴ ∠EOF = ∠ERF = 90°

Hence, PQRS is a square.

Hence proved.