P, Q, R and S are respectively the mid-points of sides AB, BC, CD and DA of quadrilateral ABCD in which AC = BD and AC ^ BD. Prove that PQRS is a square.

Question

Philoid · Accepted Answer

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

Also, AC = BD and AC is perpendicular to BD.

In ΔADC, by mid-point theorem,

SR||AC and SR = AC

In ΔABC, by mid-point theorem,

PQ||AC and PQ = AC

PO||SR and PQ = SR = AC

Now, in ΔABD, by mid-point theorem,

SP||BD and SP = BD = AC

In ΔBCD, by mid-point theorem,

RQ||BD and RQ = BD = AC

SP = RQ = AC

PQ = SR = SP = RQ

Thus, all four sides are equal.

Now, in quadrilateral EOFR,

OE||FR, OF||ER

∠EOF = ∠ERF = 90(Opposite angles of parallelogram)

∠QRS = 90

Hence, PQRS is a square.