Question

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.

Answer 1

According to the question,

We have,

P is the mid-point of the sides AB

Q is the mid-point of the sides BC

R is the mid-point of the sides CD

S is the mid-point of the sides DA

Also,

AC ⊥ BD

∠COD = ∠AOD = ∠AOB = ∠COB = 90^o

In ΔADC, by mid-point theorem,

SR = ½ AC

And, SR||AC

In ΔABC, by mid-point theorem,

PQ = ½ AC

And, PQ||AC

So, we have,

PQ||SR and SR = PQ = ½ AC

Similarly,

SP||RQ and SP = RQ = ½ BD

Now, in quadrilateral EOFR,

OE||FR and OF||ER

So, we get,

∠EOF = ∠ERF = 90^o

Hence, PQRS is a rectangle.