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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 parallelogram

we know that the diagonals of parallelogram $$PQRS$$ bisect each other.

Thus $$PR$$ and $$QS$$ bisect each other.

720089_570455_ans_7dbba3504ed24103b08a4e43823313f5.png

Mathematics

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