In the adjoining figure, ABCD is a trapezium in which AB||DC and AD = BC. If P, Q, R, S be respectively the midpoints of BA, BD, CD, CA then show that PQRS is a rhombus.

Explanation

In `triangle BDC`, Q and R are the midpoints of BD and CD respectively.
`therefore " QR||BC and " PS=(1)/(2)BC.`
Similarly , PS||BC and `PS =(1)/(2) BC.`
`therefore " PS||QR and "PS=QR " " ( "each equal to " (1)/(2) BC).`
`therefore ` PQRS is a parallelogram.
In `triangle ACD`,S and R are the midpoints of AC and CD respectively.
`therefore " SR||AD and " SR =(1)/(2) AD=(1)/(2)BC " "[ because AD =BC].`
`therefore PS = QR=SR=PQ.`
Hence, PQRS is a rhombus.