Question

Show that linesegment joining the mid points of the opposite sides of a quadrilateral bisect each other

Answer 1

In ΔADC, S and R are the midpoints of AD and DC respectively.

Recall that the line segment joining the midpoints of any two sides of a triangle is parallel to the third side and half of it.

Hence SR || AC and SR = (1/2) AC  → (1)

Similarly, in ΔABC, P and Q are midpoints of AB and BC respectively.

⇒ PQ || AC and PQ = (1/2) AC  → (2)  [By midpoint theorem]

From equations (1) and (2), we get

PQ || SR and PQ = SR  → (3)

Clearly, one pair of opposite sides of quadrilateral PQRS is equal and parallel.

Hence PQRS is a parallelogram

Hence the diagonals of parallelogram PQRS bisect each other.

Thus PR and QS bisect each other.