The midpoint of a line segment is the point that lies exactly halfway between the two endpoints. The midpoint formula is a simple and efficient way to calculate the coordinates of this midpoint. It states that the coordinates of the midpoint (M) of a line segment with endpoints A(x1, y1) and B(x<...
M = ((x1 + x2) / 2, (y1 + y2) / 2)
In essence, the midpoint's x-coordinate is the average of the x-coordinates of the endpoints, and its y-coordinate is the average of the y-coordinates of the endpoints.
Applying the Midpoint Formula to Find a Missing Endpoint
In the given problem, we know the coordinates of the midpoint M (4, 1) and one endpoint A (1, 3). We need to find the coordinates of the other endpoint B (x2, y2).
Let's use the midpoint formula and substitute the known values:
(4, 1) = ((1 + x2) / 2, (3 + y2) / 2)
Now we have two equations:
4 = (1 + x2) / 2
1 = (3 + y2) / 2
Solving for x2 and y2:
8 = 1 + x2
x2 = 7
2 = 3 + y2
y2 = -1
Therefore, the coordinates of point B are (7, -1).
Visualizing the Solution
To visualize this solution, imagine a line segment drawn on a coordinate plane with point A at (1, 3) and the midpoint M at (4, 1). The midpoint is exactly in the middle of the line segment, so to find point B, we move an equal distance from M in the opposite direction. This means moving 3 units to the right (7 - 4 = 3) and 2 units down (1 - 3 = -2), leading us to the coordinates (7, -1) for point B.
Key Takeaways
This problem demonstrates a practical application of the midpoint formula. By understanding and applying the formula, we can efficiently find the coordinates of a missing endpoint given the coordinates of the midpoint and one endpoint. Furthermore, visualizing the solution on a coordinate plane can aid in understanding the geometric relationship between the midpoint and the endpoints of a line segment.