Question

5. P Q R and S are respectively the mid-points of sides AB, BC, CD and DA of
quadrilateral ABCD in which AC = BD and AC 1 BD. Prove that PQRS is a
square

Answer 1

Answer:

Step-by-step explanation: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.

Question from  Class 9  Chapter Quadrilaterals

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Answer Text

Solution :

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∣∣ACandSR=12AC ...(i)

In ΔABC, P and Q are t he mid-points of AB and BC respectively. Then, by mid-point theorem

PQ∣∣ACandPQ=12AC ...(ii)

From Eqs. (i) and (ii),  SR=PQ=12AC ...(iii)

Similarly, in ΔBCD,  RQ∣∣BDandRQ=12BD ...(iv)

And in ΔBAD,  SP∣∣BDandSP=12BD ...(v)

From Eqs. (iv) and (v),  SP=RQ=12BD=12AC  [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.

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