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

Question

Philoid · Accepted Answer

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

Also,

AC is perpendicular to BD

∠COD = ∠AOD = ∠AOB = ∠COB = 90

In ΔADC, by mid-point theorem,

SR||AC and SR = AC

In ΔABC, by mid-point theorem,

PQ||AC and PQ = AC

PQ||SR and SR = PQ = AC

Similarly,

SP||RQ and SP = RQ = BD

Now, in quad EOFR,

OE||FR, OF||ER

∠EOF = ∠ERF = 90

Hence, PQRS is a rectangle.