Question

17. In the figure, P, Q, R and S are respectively
the mid-points of the sides of the
quadrilateral ABCD. Prove that is the
mid-point of both PR and QS.​

Answer 1

Answer:

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.

solution