Varignon Parallelogram of a Quadrilateral
Elementary School
Definition
The shape that you get when you take the midpoints of the sides of a quadrilateral and join them together to make a new four sided shape. Varignon's theorem tells us that this shape that ends up being formed is a parallelogram.
Worked examples
\(\)Square \( ABCD \)→\( \) midpoints \( M, N, P, Q \)→\( \) Varignon parallelogram \( MNPQ \) (a rhombus)\(\)
Connecting the midpoints of a square's sides forms a rhombus, which is a special parallelogram.
\(\)Rectangle \( WXYZ \)→\( \) midpoints \( E, F, G, H \)→\( \) Varignon parallelogram \( EFGH \) (also a rhombus)\(\)
For any rectangle, the Varignon parallelogram is always a rhombus with diagonals parallel to the rectangle's sides.
\(\)Trapezoid \( JKLM \)→\( \) midpoints \( R, S, T, U \)→\( \) Varignon parallelogram \( RSTU\)
Even for an irregular quadrilateral like a trapezoid, joining midpoints always produces a parallelogram.
Common mistakes
- The Varignon parallelogram is only formed when the original quadrilateral is convex. → The Varignon parallelogram forms for any quadrilateral, convex or concave. Varignon's theorem holds for all quadrilaterals, including concave and self-intersecting ones.
- Connecting any four points on the sides creates a Varignon parallelogram. → You must connect the midpoints specifically, not arbitrary points. Only the midpoints guarantee a parallelogram; other points produce unpredictable shapes.
- The Varignon parallelogram's area equals the original quadrilateral's area. → The Varignon parallelogram's area is exactly half the original quadrilateral's area. The midpoint construction always halves the area, a key property of Varignon's theorem.
Where you'll use it next
You'll use Varignon's theorem in coordinate geometry proofs, when studying properties of quadrilaterals and parallelograms, and in advanced topics like vectors and affine transformations in linear algebra.
Found in 1 StudyPug lesson
Classifying Quadrilaterals: From Basics to Advanced Concepts
5th Grade5thGrade 5 Math placeholder
Dive into the world of quadrilaterals! Discover how to identify and classify various 4-sided shapes, understand their unique properties, and apply your knowledge to real-world scenarios.
See also
Reviewed by Pat Cheng, M.Ed. — StudyPug Curriculum Lead · Last updated June 6, 2026