The quadrilateral formed by the mid-points of a quadrilateral

Objective:

To show that the figure obtained by joining the mid-points of consecutive sides of the quadrilateral is a parallelogram.

Parallelogram:

  1. A parallelogram is a simple quadrilateral with two pairs of parallel sides.
  2. The opposite or facing sides of a parallelogram are of equal length.

Proof:

In quadrilateral ABCD points P, Q, R, S are midpoints of side AB, BC, CD and AD respectively.

To prove :

PS || QR  and SR || PQ. i.e. Quadrilateral PQRS is a parallelogram


Proof:

  1. Draw diagonal BD.
  2. As PS is the midsegment of ▲ ABD, we can say that PS || BD.
  3. As QR is the midsegment of ▲ BCD, we can say that QR || BD.
  4. ∵ PS || BD and QR || BD by transitivity, we can say that PS || QR.
  5. Now draw diagonal AC.
  6. As SR is the midsegment of ▲ ACD, we can say that SR || AC.
  7. As PQ is the midsegment of ▲ ABC, we can say that PQ || AC.
  8. ∵ SR || AC and PQ || AC by transitivity, we can say that SR || PQ.
  9. ∵ PS || QR and SR || PQ, ∴ quadrilateral PQRS is a parallelogram (by definition).

 

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