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

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

Also, AC = BD

To prove: PQRS is a rhombus.

Proof: In ΔADC, S and R are the mid-points of AD and DC respectively.

Then, by mid-point theorem.

SR || AC and SR = `1/2` AC  ...(i)

In ΔABC, P and Q are the mid-points of AB and BC respectively.

Then, by mid-point theorem.

PQ || AC and PQ = `1/2` AC  ...(ii)

From equations (i) and (ii),

SR = PQ = `1/2` AC  ...(iii)

Similarly, in ΔBCD,

RQ || BD and RQ = `1/2` BD  ...(iv)

And in ΔBAD,

SP || BD and SP = `1/2` BD  ...(v)

From equations (iv) and (v),

SP = RQ = `1/2` BD = `1/2` AC  [Given, AC = BD] ...(vi)

From equations (iii) and (vi),

SR = PQ = SP = RQ

It shows that all sides of a quadrilateral PQRS are equal.

Hence, PQRS is a rhombus.