In Exercises 19–30, find the midpoint of each line segment with the given endpoints. (8, 3√5) and (−6, 7√5)

Verified step by step guidance

1. Identify the formula for the midpoint of a line segment, which is given by \( \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \).
2. Substitute the given endpoints \((x_1, y_1) = (8, 3\sqrt{5})\) and \((x_2, y_2) = (-6, 7\sqrt{5})\) into the formula.
3. Calculate the x-coordinate of the midpoint: \( \frac{8 + (-6)}{2} \).
4. Calculate the y-coordinate of the midpoint: \( \frac{3\sqrt{5} + 7\sqrt{5}}{2} \).
5. Combine the results to express the midpoint as an ordered pair.

Key Concepts

Midpoint Formula

The midpoint of a line segment in a coordinate plane is calculated using the midpoint formula, which is given by M = ((x1 + x2)/2, (y1 + y2)/2). This formula averages the x-coordinates and the y-coordinates of the endpoints to find the point that is exactly halfway between them.

Coordinate Plane

A coordinate plane is a two-dimensional surface defined by a horizontal axis (x-axis) and a vertical axis (y-axis). Each point on the plane is represented by an ordered pair (x, y), where 'x' indicates the horizontal position and 'y' indicates the vertical position. Understanding how to plot points and interpret their coordinates is essential for solving problems involving line segments.

Endpoints of a Line Segment

Endpoints are the two points that define a line segment in a coordinate plane. In this context, the endpoints are given as (8, 3√5) and (−6, 7√5). Recognizing these points is crucial for applying the midpoint formula correctly, as they provide the necessary coordinates to calculate the midpoint.