Medians and altitudes are both important segments in a triangle, but they serve different purposes and have distinct definitions.
Medians: A median of a triangle is a line segment that connects a vertex to the midpoint of the opposite side. Every triangle has three medians, and they all meet at a point called the centroid. The centroid is the center of mass of the triangle, and it divides each median into two segments that are in a 2:1 ratio. This means that the segment from the centroid to the midpoint is one-third the length of the segment from the vertex to the centroid.
Altitudes: An altitude, on the other hand, is a line segment drawn from a vertex perpendicular to the line containing the opposite side. This means that the altitude represents the height of the triangle from that vertex. Each triangle also has three altitudes, and they intersect at a point called the orthocenter. The orthocenter’s position varies depending on the type of triangle: it lies inside the triangle for acute triangles, on the triangle for right triangles, and outside for obtuse triangles.
In summary, while both medians and altitudes connect vertices to other parts of the triangle, medians connect vertices to midpoints, and altitudes connect vertices to opposite sides at right angles. Understanding the difference between these two elements is essential in geometry, as they have various applications in solving problems and analyzing triangle properties.