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

2 thoughts on “Show that the line segments joining the mid-points of the opposite sides of a quadrilateral bisect each other​”

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!!]

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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 ☺️

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