What does the median of a triangle connect?

The median of a triangle is defined as a line segment that connects one vertex of the triangle to the midpoint of the side opposite that vertex. This means that for each triangle, there are three medians, each corresponding to one of the triangle's vertices.

When a median is drawn, it divides the triangle into two smaller triangles of equal area. This property is significant in understanding various concepts within geometry, including the balance point of a triangle, which is often represented by the centroid, the point where all medians intersect.

The other options reference different segments or connections within a triangle. While the centroid is the point where the medians intersect, the statement about a vertex to the centroid does not accurately describe what a median is. Likewise, connecting the midpoints of two sides doesn’t relate to the concept of a median; those segments are called midsegments. Finally, connecting all three vertices of the triangle describes the triangle itself, not a median.

Therefore, the correct answer illustrates the specific relationship of the median to the triangle's structure, emphasizing its role in connecting a vertex directly to the midpoint of the opposite side.