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Geometry

Students derive a formula for the coordinates of the centroid of a triangle and use the formula to find the coordinates of the centroid when provided with the coordinates of the triangle’s vertices.

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After completing this tutorial, you will be able to complete the following:

- After completing this Activity Object, students will be able to:
- Know that the coordinates of the centroid of a triangle are an average of the vertex coordinates.
- Prove that the coordinates of the centroid of a triangle are averages of the vertex coordinates.
- Find the coordinates of the centroid of a triangle given the coordinates of the triangle vertices.

The median of a triangle is a line segment from a vertex of a triangle to the midpoint of the opposite side. Each median of a triangle divides the triangle into two triangles with equal area. A triangle can be divided into three triangles with equal area by drawing all three medians. In the areas of triangles AGB, BGC and CGA are equal.

The three medians of a triangle intersect at a point known as the centroid of the triangle. In , G is the centroid of the triangle. If a triangle is made out of a uniformly dense material, then the centroid is the center of mass of the triangle. For example, if the triangle is made out of a piece of uniformly dense paper, it will balance perfectly on top of a pin if the pin is located at the centroid. Informally, it is the average of all the points in the triangle. In fact, the coordinates of the centroid are the averages of the corresponding coordinates of the vertices of the triangle. In order to prove this, we need the following fact about how the centroid cuts the medians:

The centroid of the triangle cuts the medians in a constant ratio. It is the point on all three medians that is of the way from a given vertex to the opposite midpoint and of the way from the midpoint of the side to the opposite vertex. In other words, since AD is a median and G is the centroid of ABC,

The proof that the coordinates of the centroid are the arithmetic mean of the corresponding coordinates of the vertices of the triangle also relies on vectors. Recall that a quantity is a measurable attribute of an object. The size of a quantity is known as its magnitude. Some quantities consist solely of magnitude and are known as scalars. Other quantities, called vectors, consist of a magnitude and a direction. Formally, vector quantities are denoted with an arrow overhead (such as They are represented by a vector with a directed line segment, or a line segment with one endpoint identified as the initial point, the other identified as the terminal point-thought to be directed from the initial point to the terminal point.

The length and direction of the directed line segment correspond to the magnitude and direction of the vector. Note that vectors do not have an initial or a terminal point, only a direction and a magnitude. However, we can represent vectors with directed line segments that do have initial and terminal points. Thus, different directed line segments can represent the same vector.

We can also represent vectors in the plane with coordinates. The horizontal (or x-) coordinate of a vector is the horizontal distance from the initial point to the terminal point of any directed line segment representing the vector. Similarly, the vertical (or y-) coordinate of a vector is the vertical distance from the initial point to the terminal point of any directed line segment representing the vector. The vector's coordinates indicate the direction and magnitude of the vector. The vector with coordinates (2, 3) is represented by any directed line segment that has its terminal point two units to the right and three units up from its initial point. Notice that when we place the initial point of the directed line segment representing at the origin, the vector's coordinates are exactly the coordinates of the terminal point:

Note that vector also has vector coordinates (2, 3), since it is represented by a directed line segment that has its terminal point two units to the right and three units up from its initial point. However, the directed line segment representing has initial point (-5, -2), and terminal point (-3, 1).

We can use vector coordinates to find the coordinates of the centroid in a triangle.

Approximate Time | 20 Minutes |

Pre-requisite Concepts | Students should be able to define the midpoint of a line segment, determine the coordinates of the midpoint of a line segment, define of the centroid and medians in a triangle, and state how the centroid divides the medians. |

Course | Geometry |

Type of Tutorial | Skills Application |

Key Vocabulary | centroid, coordinates of the centroid, vertices |