Question

Prove that the figure obtained by joining the mid-points of the adjacent sides of a rectangle is a rhombus.

Answer 1

Join AC and BC.
In ΔABC, P and Q are the mid-point of AB and BC respectively.

PQ = `(1)/(2)"AC"`.......(i) and PQ || AC

In ΔBDC, R and Q are the mid-points of CD and BC respectively.

QR = `(1)/(2)"BD"`.......(ii) and QR || BD

But AC = BD  ...(diagonals of a rectangle)
From (i) and (ii)
PQ = QR
Similarly, QR = RS, RS = SP and RS || AC, SP || BD
Hence, PQ = QR = PS = SP
Therefore, PQRS is a rhombus.