The gravicentro is a definition that is very used in geometry when working with triangles.
To understand the definition of gravity, it is first necessary to know the definition of "medians" of a triangle.
The medians of a triangle are the line segments that start at each vertex and reach the midpoint of the side opposite that vertex.
The point of intersection of the three medians of a triangle is called the barycenter or it is also known as the gravicenter.
It is not enough just to know the definition, it is interesting to know how this point is calculated.
Calculation of the center of gravity
Given a triangle ABC with vertices A = (x1, y1), B = (x2, y2) and C = (x3, y3), the gravicenter is the intersection of the three medians of the triangle.
A quick formula that allows the calculation of the gravity center of a triangle, being known the coordinates of its vertices is:
G = ((x1 + x2 + x3) / 3, (y1 + y2 + y3) / 3).
With this formula, you can find out the location of the graviccenter in the Cartesian plane.
Characteristics of the Gravicentro
It is not necessary to draw the three medians of the triangle, because when drawing two of them, it will be evident where the gravicentro is.
The gravicentro divides each median into 2 parts whose proportion is 2: 1, that is, the two segments of each median are divided into segments of lengths 2/3 and 1/3 of the total length, the greater distance being the one there is between the vertex and the gravity center.
The following image better illustrates this property.
The formula for calculating gravity is very simple to apply. The way to obtain this formula is by calculating the line equations that define each median and then finding the intersection point of these lines.
Exercises
Here is a short list of problems about calculating the center of gravity.
1.- Given a triangle with vertices A = (0,0), B = (1,0) and C = (1,1), calculate the gravity center of said triangle.
Using the given formula, it can be quickly concluded that the gravity center of triangle ABC is:
G = ((0 + 1 + 1) / 3, (0 + 0 + 1) / 3) = (2/3, 1/3).
2.- If a triangle has vertices A = (0,0), B = (1,0) and C = (1 / 2,1), what are the coordinates of the gravicentro?
Since the vertices of the triangle are known, we proceed to apply the formula for calculating the gravity center. Therefore, the gravicentro has coordinates:
G = ((0 + 1 + 1/2) / 3, (0 + 0 + 1) / 3) = (1/2, 1/3).
3.- Calculate the possible gravicentros for an equilateral triangle such that two of its vertices are A = (0,0) and B = (2,0).
In this exercise you are only specifying two vertices of the triangle. In order to find the possible gravicentros, we must first calculate the third vertex of the triangle.
Since the triangle is equilateral and the distance between A and B is 2, the third vertex C must be at distance 2 from A and B.
Using the fact that in an equilateral triangle the height coincides with the median and also using the Pythagorean theorem, it can be concluded that the options for the coordinates of the third vertex are C1 = (1, √3) or C2 = (1, - √3).
So the coordinates of the two possible gravicentries are:
G1 = ((0 + 2 + 1) / 3, (0 + 0 + √3) / 3) = (3/3, √3 / 3) = (1, √3 / 3), G2 = ((0 + 2 + 1) / 3, (0 + 0-√3) / 3) = (3/3, -√3 / 3) = (1, -√3 / 3).
Thanks to the previous accounts, it can also be noted that the median was divided into two parts whose proportion is 2: 1.
References
- Landaverde, F. d. (1997). Geometry (Reprint ed.). Progress.
- Leake, D. (2006). Triangles (illustrated ed.). Heinemann-Raintree.
- Pérez, CD (2006). Precalculation. Pearson Education.
- Ruiz, Á., & Barrantes, H. (2006). Geometries. CR technology.
- Sullivan, M. (1997). Precalculation. Pearson Education.
- Sullivan, M. (1997). Trigonometry and Analytical Geometry. Pearson Education.