show that the line segment joining the midpoint of the opposite side of a quadrilateral bisect each other
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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
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
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