Question

Show that the line segment joining the mid point of the opposite sides of a quadrilateral bisect each other

Answer 1

Answer:

Given: ABCD is a quadrilateral in which PR and SQ are the line segments joining the mid-points of the opposite sides.

To prove: PR and SQ bisect each other. Construction: Join PQ, QR, RS, SP and AC.

Proof:

In ΔABC, P and Q are mid-points of sides AB and BC respectively.

From mid-point theorem

PQ || AC and PQ =  1/2  AC …(i)

In ΔADC, we have

S and R are mid-points of sides AD and CD respectively.

Therefore, from mid-point theorem

SR || AC and SR =  1/2  

AC …(ii) From (i) and (ii),

we get PQ || SR and PQ = SR ⇒ PQRS is a parallelogram.

Reason: A quadrilateral is a parallelogram if its one pair of opposite sides is equal and parallel.

Now PR and SQ are diagonals of the parallelogram PQRS. PR and SQ bisect each other.

(∵ Diagonals of a parallelogram bisect each other)