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

Question

Philoid · Accepted 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.