What is geometry is a median?

In geometry, a median is a line segment that joins a vertex of a triangle to the midpoint of the opposite side. This bisects that side of the triangle and divides it into two equal parts. 

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Medians are found in all types of triangles, including isosceles and equilateral triangles. They are also found in tetrahedra and trapezoid. 

Identifying the medians of triangles

Each triangle has three medians, one from each vertex, that all intersect at a point called the centroid. The centroid is the “balancing point” for the triangle and is the center of gravity for the entire triangle. The medians are the lines that connect each of the three vertices to the centroid and that are also used to measure properties like perimeter and area of the triangle. 

Calculating the length of a median 

In a triangle, the median is the line segment that connects a vertex to the midpoint of the opposite side. The median can be found by using a formula that calculates the length of the sides, the midpoint of the opposite side, and the median. 

Finding the length of a median is important to know for geometry and science because it is used to calculate areas and perimeters of triangles and other shapes. The medians of a triangle are also used to find the distance from each vertex to the centroid. 

Using the geometric median

The geometric median is a procedure that can be used to approximate the geometric median of a set of discrete points. It is based on the fact that each of the sample points has its own Euclidean distance from the other vertices, and that the sum of the distances to all the points is a convex function. 

Once the distance to each of the points is computed, the geometric median can be calculated by using an iterative procedure that increases the sum of each of the distances at every step. The iterative process cannot get trapped in a local optimum, which makes it an efficient method of approximating the geometric median. 

Using the geometric median with an interactive tool

Students can use the interactive tools on the TracenPoche website to explore the relationship between the parts of a median and the centroid of the triangle. They can drag the vertices of the triangle and then watch the ratios display at the bottom of the screen. 

They can then explore the relationship between each part of the median and the centroid by dragging them around again. They will then see that each part of the median is divided into two pieces by the centroid and that one of the pieces is twice as long as the other piece. 

They can then learn to draw or identify medians in triangles, use the geometric median to determine the centroid of a triangle, and calculate the length of a median. They can also discover that a triangle with three medians has a geometric median.