Question

Question 3
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.

Thinking Process

Firstly, use the mid-point theorem in various triangles of a quadrilateral. Further show that the line segments formed by joining the mid-points are equal, which prove the required quadrilateral.

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 $\Delta ADC$, S and R are the mid-points of AD and DC respectively. Then, by mid-point theorem.

SR|| AC and $SR=\frac{1}{2}$ AC …………………………(i)

In $\Delta ABC,P$ and Q are the mid-points of AB and BC respectively. Then, by mid-point theorem

PQ||AC and $PQ=\frac{1}{2}AC$ …………………(ii)

From eqs. (i) and (ii), $SR=PQ=\frac{1}{2}AC$ …………..(iii)

Similarly, in $\Delta BCD$, RQ||BD and RQ = $\frac{1}{2}$ BD ………………..(iv)

And in $\Delta BAD$, SP||BD and $SP=\frac{1}{2}BD$ ……………………(v)

From eqs. (iv) and (v), $SP=RQ=\frac{1}{2}BD=\frac{1}{2}AC$ [given, AC=BD] …………………(vi)

From eqs. (iii) and (vi), SR = PQ = SP = RQ

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

Hence, PQRS is a rhombus. Hence proved.