Question

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

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, we know that,

AC = BD.

In ΔADC, by mid-point theorem,

SR = ½ AC

And, SR||AC

In ΔABC, by mid-point theorem,

PQ = ½ AC

And, PQ||AC

Hence, SR = PQ = ½ AC

Similarly,

In ΔBCD, by mid-point theorem,

RQ = ½ BD

And, RQ||BD

In ΔBAD, by mid-point theorem,

SP = ½ BD

And, SP || BD

So, we get,

SP = RQ = ½ BD = ½ AC

Then,

SR = PQ = SP = RQ

Hence, PQRS is a rhombus.