Question

Show that the line segment joining the mid-points of the opposite sides of a quadrilateral bisect each other.​

Answer 1

ABCD is a quadrilateral in which P, Q, R, & S are mid points of AB, BC, CD & AD

In Δ ACD

SR is touching mid points of CD and AD

So, SR || AC

Similarly following can be proved

PQ || AC

QR || BD

PS || BD

So, PQRS is a parallelogram.

PR and QS are diagonals of the parallelogram PQRS, so they will bisect each other.

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Answer 2

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.