Question

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

Answer 1

Given:-

ABCD is a quadrilateral P,Q,R & S are the midpoints of the respective sides.

To Prove:-

PR and QS bisect each other

Proof:-

By midpoint Theorem:-

 Join PQ,QR,RS,PS

Join diagonals AC and BD

• In ΔABC,

→P and Q r the midpoints of AB and BC respectively

→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

[Hence Proved!!]

Answer 2

ABCD is a quadrilateral such that P, Q, R and s are the mid points of sides AB, BC, CD and DA respectively. (see the image)

In ABC, P and Q are the mid points of sides AB and BC respectively.

• Therefore, PQ || AC and PQ = 1/2 of AC

• Similarly, RS || AC and RS = 1/2 of AC

• PQ || RS and PQ = RS

• Similarly, PQ || QR and PQ = QR

Hence, PQRS is a parallelogram.

Since the diagonals of a parallelogram bisect each other,

PR and QS bisect each other.

HOPE this helps you ☺️