ABCD is a quadrilateral in which AD=BC. If P,Q,R,S are the midpoints of AB, AC, CD and BD respectively, show that PQRS is a rhombus

Draw a quadrilateral ABCD with AD = BC and join AC, BD. P, Q, R, S are the mid points of AB, AC, CD and BD respectively.

Now, In triangle ABC, P and Q are mid points of AB and AC respectively.

So, PQ || BC and PQ = BC ..... (1)

Similarly in ΔADC, QR = AD = BC .... (2)

Now consider ΔBCD,

SR = BC ...... (3)

Similarly, in ΔABD,

PS = AD = BC ...... (4)

∴ From (1), (2), (3) and (4), we get –

PQ = QR = SR = PS

Since all sides are equal, so PQRS is a rhombus.

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