Question

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.

Answer 1

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 = $\frac{1}{2}$BC ..... (1)

Similarly in ΔADC, QR = $\frac{1}{2}$AD = $\frac{1}{2}$BC .... (2)

Now consider ΔBCD,

SR = $\frac{1}{2}$BC ...... (3)

Similarly, in ΔABD,

PS = $\frac{1}{2}$AD = $\frac{1}{2}$BC ...... (4)

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

PQ = QR = SR = PS

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