Question

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

Answer 1

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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

Answer:

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.

Step-by-step explanation: