Question

Abcd is a quadrilateral in which ad =bc. If p, q, r, s be the mid point of ab, ac, cd, bd respectively, than show that pqrs is a rhombus

Answer 1

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.