What point is determined by averaging the coordinates of a segment's endpoints?

The correct answer is the midpoint, which is a specific point that is determined by averaging the coordinates of the endpoints of a line segment. To find the midpoint of a segment defined by two endpoints, you take the x-coordinates of the endpoints, add them together, and divide by 2. You do the same with the y-coordinates. Mathematically, if you have endpoints ((x_1, y_1)) and ((x_2, y_2)), the coordinates of the midpoint (M) can be calculated as:

[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)

]

This method ensures that the midpoint lies exactly halfway between the two points on the segment, making it a fundamental concept in geometry for dividing segments into equal lengths.

Other options represent different concepts: the centroid refers to the average position of all points in a shape, often used in the context of triangles; intersection points result from the crossing of two or more lines or curves; and circumference is a term related to the distance around a circle, not applicable for finding specific points along a line segment.