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

GIVEN:

• ABCD is a quadrilateral

• AD = BC

• P, Q, R, S are mid points of AB, BD, CD, and AC respectively.

TO PROVE:

Prove that PQRS is a rhombus.

PROOF:

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,

• PS || AD and PS = 1/2 AD......➊

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

So , By mid-point theorem,

• OR || AD and QR = 1/2 AD........➋

Compare ( 1 ) and ( 2 ), we get

PS || QR and PS = QR

Since, one pair of opposite sides is equal as well as parallel then,

PQRS is a parallelogram.......... ➌

NOW,

In ∆ ABC , by mid-point theorem

• PQ || BC and PQ = 1/2 BC..........➍

• and AD = BC .........➎

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

PS = PQ

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

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