Medians and Altitudes of Triangles
This page introduces the concepts of medians and altitudes in triangles, along with their properties and the theorems related to their concurrency.
Definition: A median is a segment that connects a vertex of a triangle to the midpoint of the opposite side.
Highlight: The point where the medians of a triangle intersect is called the centroid, also known as the center of gravity.
The Concurrency of Medians Theorem states that the medians of a triangle intersect at a point that is two-thirds of the distance from each vertex to the midpoint of the opposite side.
Example: In a triangle where XA = 8, the length of XB can be calculated as XB = XA + AB = 8 + 4 = 12.
Definition: An altitude of a triangle is a perpendicular segment from a vertex to the line containing the opposite side.
Altitudes can be inside, outside, or coincide with a side of the triangle, depending on the triangle's shape.
The Concurrency of Altitudes Theorem states that the lines containing the altitudes of a triangle are concurrent, meaning they intersect at a single point.
Vocabulary: The point where the altitudes intersect is called the orthocenter of the triangle.
