10th Grade - Barycenter

10th Grade

7 - Barycenters

Definition of the barycenter

Take a triangular sheet and put on it, at point A, a 1 kg weight, at point B, a 2kg weight, and at point C a 3kg weight. The barycenter of the system  ponderates points is the centre of gravity of the sheet, that means the point where the sheet is balanced. Call it G. You guess that G will be closer to C than to A. G verifies the vector equality : propriete barycenter
In general, the barycenter G of ponderates points verifies :

propriete fondamentale du barycenter

How to build a barycenter

To draw it, decompose 2 of the 3 vectors according to the vectors you can already build. Long but simple :
calculation construction barycenter

So G is here :
vecteurs triangle

Fundamental property of the barycenter

You've got to know that if G is the barycenter of the system ponderates points, then for every point M in the plane, we have :
propriete du barycenter

The drawing below with the figures at the beginning illustrates it well.

dessin de construction du barycenter
In practice and in problems you will usually have to place M at a particular place to prove things.

Coordinates of a barycenter

If A, B, and C are 3 points in an orthonormal frame, with coordonnées points, and if G is the barycenter of  ponderates points, then the folowing formulas give the coordinates of G :

coordonnées barycenter

You can use the formulas in case there are more than 3 points or in simple cases, with only 2 points. Finally you have to know that the barycenter of a system of points that all have same coefficients (the same weights) is called the isobarycenter.

>>> Lesson on the dot product >>>

Barycenters in 10th Grade

lesson, problems