In Exercises 105–106, find the midpoint of each line segment with the given endpoints. (2, 6) and (-12, 4)

Verified step by step guidance

1. Identify the formula for the midpoint of a line segment: $M=(\frac{x+x\text{'}}{2},\frac{y+y\text{'}}{2})$, where (x, y) and (x', y') are the endpoints of the line segment.
2. Substitute the given coordinates of the endpoints into the formula. Here, (x, y) = (2, 6) and (x', y') = (-12, 4).
3. Calculate the x-coordinate of the midpoint: $\frac{2+(-12)}{2}$.
4. Calculate the y-coordinate of the midpoint: $\frac{6+4}{2}$.
5. Combine the calculated x-coordinate and y-coordinate to express the midpoint as an ordered pair: $(x,y)$.

Key Concepts

Midpoint Formula

The midpoint formula is used to find the midpoint of a line segment defined by two endpoints in a coordinate plane. It is given by the formula M = ((x1 + x2)/2, (y1 + y2)/2), where (x1, y1) and (x2, y2) are the coordinates of the endpoints. This formula calculates the average of the x-coordinates and the average of the y-coordinates, resulting in the coordinates of the midpoint.

Coordinate Plane

The coordinate plane is a two-dimensional surface formed by the intersection of a horizontal axis (x-axis) and a vertical axis (y-axis). Each point on this plane is represented by an ordered pair (x, y), where 'x' indicates the horizontal position and 'y' indicates the vertical position. Understanding the coordinate plane is essential for visualizing and solving problems involving points and line segments.

Endpoints of a Line Segment

Endpoints of a line segment are the two distinct points that define the segment. In the context of the given question, the endpoints are (2, 6) and (-12, 4). These points are crucial for applying the midpoint formula, as they provide the necessary coordinates to calculate the midpoint, which represents the point that is equidistant from both endpoints.