Question

# Show that the line segments joining the midpoints of the opposite sides of a quadrilateral and bisect each other

Solution

## Given $$ABCD$$ is quadrilateral , $$S$$ and $$R$$ are the midpoints of $$AD$$ and $$DC$$ respectively.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 $$SR\parallel AC$$ $$SR=\dfrac{1}{2}AC$$........ (1)Similarly, in $$\Delta ABC$$, $$P$$ and $$Q$$ are midpoints of $$AB$$ and $$BC$$ respectively.$$PQ\parallel AC$$ $$PQ=\dfrac{1}{2}AC$$....... (2)  [By midpoint theorem]From equations (1) and (2), we get$$PQ\parallel SR$$ $$SR=PQ$$................(3)Clearly, one pair of opposite sides of quadrilateral $$PQRS$$ is equal and parallel.Hence $$PQRS$$ is a parallelogramwe know that the diagonals of parallelogram $$PQRS$$ bisect each other.Thus $$PR$$ and $$QS$$ bisect each other.Mathematics

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