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Searching for ## Median, Altitude, and Bisector

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Geometry

Students learn the definitions of median and centroid in a triangle, discover and prove that medians cut each other in the same ratio, and draw the medians of a triangle using that property.

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

- Define the median of a triangle.
- Define the centroid of a triangle.
- Explain the ratio of the segment lengths into which the centroid divides any median.

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. Consider triangle ABC with median AD below. Notice that triangle ABD and ACD both have AH as an altitude. Since AD is a median, |BD|=|DC|, so ABD and ACD have the same area.

A triangle can be divided into three triangles with equal area by drawing all three medians. In Figure 2, 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 Figure 1, G is the centroid of the triangle. 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. This fact is expanded upon in the "Coordinates of the Centroid of a Triangle" Activity Object. 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.

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,

This means that

Approximate Time | 20 Minutes |

Pre-requisite Concepts | Students should know how to find the area of a triangle, find the midpoint of a line segment, and be familiar with the triangle midsegment theorem. |

Course | Geometry |

Type of Tutorial | Concept Development |

Key Vocabulary | centroid, median, median of a triangle |