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. Answer:

   In △ADC,S is the mid-point of AD and R is the mid-point of CD

   In △ABC,P is the mid-point of AB and Q is the mid-point of BC

   Line segments joining the mid-points of two sides of a triangle is parallel to the third side and is half of of it.

   ∴SR∥AC and SR=

   2

   1

   AC ….(1)

   ∴PQ∥AC and PQ=

   2

   1

   AC ….(2)

   From (1) and (2)

   ⇒PQ=SR and PQ∥SR

   So,In PQRS,

   one pair of opposite sides is parallel and equal.

   Hence, PQRS is a parallelogram.

   PR and SQ are diagonals of parallelogram PQRS

   So,OP=OR and OQ=OS since diagonals of a parallelogram bisect each other.

   Hence proved.

   Step-by-step explanation:

   I HOPE THAT IT’S HELPFUL FOR YOU.

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