Show that the line segment joining the midpoint of opposite sides of a quadrilateral bisect each other
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Answers
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 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
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.