Question

Q
In the adjoining figure, P Q R and S are the mid-points of
AB, BC, CD and AD respectively. Prove that PQRS is a
parallelogram.​

Answer 1

Step-by-step explanation:

8th

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Area of Trapezium

P,Q,R,R, and S are respecti...

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Asked on January 17, 2020 by

Dipankar Manivannan

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

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ANSWER

P,Q,R and S are the mid-point of the sides AB,BC,CD and DA of a quadrilateral ABCD.

⇒ AC=BD

In △ABC,

P and Q are the mid-points of the sides AB and BC respectively.

∴ PQ∥AC ----- ( 1 )

And PQ=

2

1

×AC ------ ( 2 )

Similarly, SR∥AC and SR=

2

1

×AC ----- ( 3 )

From ( 1 ), ( 2 ) and ( 3 ) we get,

⇒ PQ∥SR and PQ=SR=

2

1

×AC ----- ( 4 )

Similarly we an show that,

⇒ SP∥RQ and SP=RQ=

2

1

×BD ----- ( 5 )

Since, AC=BD

∴ PQ=SR=SP=RQ [ From ( 4 ) and ( 5 ) ]

All sides of the quadrilateral are equal.

∴ PQRS is a rhombus.