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Home » Quadrangles » Parallelogram Formed by Connecting the Midpoints of a Quadrilateral

Parallelogram Formed by Connecting the Midpoints of a Quadrilateral

Last updated: Mar 27, 2021 by Ido Sarig · This website generates income via ads and uses cookies · Terms of use · Privacy policy

If you connect the midpoints of the sides of any quadrilateral, the resulting quadrilateral is always a parallelogram.

Surprisingly, this is true whether it is a special kind of quadrilateral like a parallelogram or kite or trapezoid, or just any arbitrary simple convex quadrilateral with no parallel or equal sides.

Problem

In a quadrilateral ABCD, the points P, Q, R and S are the midpoints of sides AB, BC, CD and DA, respectively. Prove the PQRS is a parallelogram.

quadrilateral midpoints

Strategy

The fact that we are told that P, Q, R and S are the midpoints should remind us of the Triangle Midsegment Theorem - the midsegment is parallel to the third side, and its length is equal to half the length of the third side.

We have no triangles here, so let's construct them, so the midpoints of the quadrilateral become midpoints of triangles, by drawing the diagonal AC:

quadrilateral midpoints with diagonal

We now have two triangles, ΔBAC and ΔDAC, where PQ and SR are midsegments. So, using the Triangle Midsegment Theorem we find that PQ||AC and PQ = ½AC, and also that SR||AC and SR = ½AC.

Since PQ and SR are both parallel to a third line (AC) they are parallel to each other, and we have a quadrilateral (PQRS) with two opposite sides that are parallel and equal, so it is a parallelogram.

We could have also done this by drawing the second diagonal DB, and used the two triangles ΔADB and ΔCDB instead.

Proof

This is how you show that connecting the midpoints of quadrilateral creates a parallelogram:

(1) AP=PB //Given
(2) BQ=QC //Given
(3) PQ||AC //(1), (2), Triangle midsegment theorem
(4) PQ = ½AC //(1), (2), Triangle midsegment theorem
(5) AS=SD //Given
(6) CR=RD //Given
(7) SR||AC //(5), (6), Triangle midsegment theorem
(8) SR = ½AC //(5), (6), Triangle midsegment theorem
(9) SR=PQ //(4), (8), Transitive property of equality
(10) SR||PQ //(3), (7), two lines parallel to a third are parallel to each other
(11) PQRS is a Parallelogram //Quadrilateral with two opposite sides that are parallel & equal

« Two Lines Parallel to a Third are Parallel to Each Other
Midpoints of a Quadrilateral - a Difficult Geometry Problem »

About the Author

Ido Sarig is a high-tech executive with a BSc degree in Computer Engineering. His goal is to help you develop a better way to approach and solve geometry problems. You can contact him at [email protected]

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Welcome to Geometry Help! I'm Ido Sarig, a high-tech executive with a BSc degree in Computer Engineering and an MBA degree in Management of Technology. I'm here to tell you that geometry doesn't have to be so hard! My goal with this website is to help you develop a better way to approach and solve geometry problems, even if spatial awareness is not your strongest quality. Read More…

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