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Question

In the quadrilateral given below, AD=BC.P,Q,R and S are mid-points of AB,BD,CD and AC respectively. Prove that PQRS is a rhombus.
693987_41e183873d634f26b760025751758150.png

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Solution


AD=BC [ Given ]
The mid-point theorem states that the segment joining two sides of triangle at the mid-points of those sides is parallel to the third side and is half the length of the third side.

In BAD,
P and S are the mid-points of sides AB and BD.

So, By mid-point theorem,
PSAD and PS=12AD ------- ( 1 )

In CAD,
R and Q are the mid-point of CD and AC.

So, by mid-point theorem,
ORAD and QR=12AD ----- ( 2 )

Compare ( 1 ) and ( 2 ), we get

PSQR and PS=QR
Since, one pair of opposite sides is equal as well as parallel then,
PQRS is a parallelogram ---- ( 3 )

Now, In ABC, by mid-point theorem
PQBC and PQ=12BC ----- ( 4 )

And AD=BC ----- ( 5 )

Compare equations ( 1 ), ( 4 ) and (5) we get,

PS=PQ ----- ( 6 )

Since, PQRS is a parallelogram with PS=PQ then PQRS is a rhombus.

1441603_693987_ans_5c9c00b6380f4e5ea55a50056f37c9d0.png

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