Question

Prove that the figure formed by joining the midpoints of the adjacent sides of a quadrilateral is a parallelogram.​

Answer 1

Answer:

Let ABCD be any quadrilateral.

Join A and C.

Let P and Q be midpoints of sides AB and BC

In △ABC,

PQ∥AC and PQ=

2

1

AC ............(i) (by Midpoint theorem)

S and R be the midpoints of AD and DC respectively

In △ACD,

SR∥AC and SR=

2

1

AC .............(ii) (by Midpoint theorem)

From (i) and (ii),

PQ∥SR and PQ=SR.

∵ One pair of opposite sides are parallel and equal

∴ ABCD is a parallelogram.