Definition of a Median
In geometry, a median of a triangle is a line segment that joins a vertex to the midpoint of the opposite side. Every triangle has exactly three medians, one from each vertex.
Key Properties and the Centroid
The three medians of a triangle always intersect at a single point called the centroid of the triangle. The centroid is also the triangle's center of mass, meaning if you were to cut out a triangle from a uniform material, it would balance perfectly on its centroid. Each median divides the triangle into two smaller triangles of equal area, and the centroid divides each median in a 2:1 ratio, with the longer segment being from the vertex to the centroid.
Practical Example: Finding a Median
Consider a triangle ABC with vertices A(x1, y1), B(x2, y2), and C(x3, y3). To find the median from vertex A to side BC, first find the midpoint M of side BC using the midpoint formula: M = ((x2+x3)/2, (y2+y3)/2). The median is then the line segment AM. Its length can be calculated using the distance formula between points A and M.
Importance and Applications
Medians are fundamental in understanding the internal structure and balance of triangles. They are used in various geometric proofs, problems involving area division, and in engineering applications where the center of mass of triangular components needs to be precisely located for stability and balance, such as in structural design or robotics.