Question

${\red{\small{Show that the line segment joining mid-points of the opposite sides of a quadrilateral bisects each other.}}}$​

Answer 1

Let ABCD is a quadrilateral in which P, Q, R, and S are the mid-points of sides AB, BC, CD, and DA respectively. Join PQ, QR, RS, SP, and BD.

In ΔABD, S and P are the mid-points of AD and AB respectively. Therefore, by using mid-point theorem, it can be said that

SP || BD and SP =  BD ... (1)

Similarly in ΔBCD,

QR || BD and QR = BD ... (2)

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

SP || QR and SP = QR

In quadrilateral SPQR, one pair of opposite sides is equal and parallel to

each other. Therefore, SPQR is a parallelogram.

We know that diagonals of a parallelogram bisect each other.

Hence, PR and QS bisect each other.


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Answer 2

$\small\bf\red{\texmathcal{Given\: : \:ABCD\: is\: a \:quadrilateral}}$

           P,Q,R&S r the midpoints of the respective sides

To prove:PR and QS bisect each other

Proof:

 Join PQ,QR,RS,PS

Join diagonals AC and BD

In ΔABC,

P and Q r the midpoints of AB and BC respective

Therefore by midpoint theorem, PQ is parallel to AC and PQ=1/2AC

In the same way prove that SR is parallel to AC and SR=1/2AC

Therefore since the opposite sides are equal and parallel PQRS is a parallelogram

In a parallelogram diagonals bisect each other

$\huge\pink{\mathcal{Hence \:Proved}}$