ABCD is a quadrilateral in which AD = BC. If P, Q, R, S be 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 midpoints of AB, AC, CD and BD respectively.
Now, In triangle ABC, P and Q are midpoints of AB and AC respectively.
So, PQ || BC and PQ = 12BC ..... (1)
Similarly in ΔADC, QR = 12AD = 12BC .... (2)
Now consider ΔBCD,
SR = 12BC ...... (3)
Similarly, in ΔABD,
PS = 12AD = 12BC ...... (4)
∴ From (1), (2), (3) and (4), we get –
PQ = QR = SR = PS
Since all sides are equal, so PQRS is a rhombus.