Determining the location of a line segment's endpoint when given one endpoint and its midpoint is a fundamental concept in geometry and coordinate geometry. This process relies on the midpoint formula, which expresses the relationship between the coordinates of the endpoints and the midpoint of a line segment. This article will delve into the process of finding the unknown endpoint of a line segment when you're provided with the coordinates of one endpoint and the midpoint.
Understanding the Midpoint Formula
The midpoint formula is a cornerstone in coordinate geometry. It states that the coordinates of the midpoint of a line segment are the average of the x-coordinates and the average of the y-coordinates of the endpoints. Mathematically, if the endpoints of a line segment are (x1, y1) and (x2, y2), then the midpoint (xm, ym) can be calculated as:
- xm = (x1 + x2) / 2
- ym = (y1 + y2) / 2
Finding the Unknown Endpoint
Let's consider a scenario where you are given one endpoint (x1, y1) and the midpoint (xm, ym) of a line segment, and you need to determine the coordinates of the other endpoint (x2, y2).
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The Given Information: You are given the coordinates of one endpoint, (x1, y1), and the midpoint, (xm, ym).
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Applying the Midpoint Formula: The midpoint formula provides the foundation for solving this problem. We can substitute the known values into the midpoint formula and solve for the unknown endpoint coordinates.
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Solving for x2: We can rearrange the midpoint formula for x to solve for x2:
- xm = (x1 + x2) / 2
- 2 * xm = x1 + x2
- x2 = 2 * xm - x1
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Solving for y2: Similarly, we can rearrange the midpoint formula for y to solve for y2:
- ym = (y1 + y2) / 2
- 2 * ym = y1 + y2
- y2 = 2 * ym - y1
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The Unknown Endpoint: The calculated values of x2 and y2 represent the coordinates of the unknown endpoint of the line segment.
Illustrative Example
Let's illustrate this with a concrete example. Suppose one endpoint of a line segment is at (2, 5) and the midpoint is at (4, 3). We want to find the other endpoint.
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Given:
- (x1, y1) = (2, 5)
- (xm, ym) = (4, 3)
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Solving for x2:
- x2 = 2 * xm - x1
- x2 = 2 * 4 - 2
- x2 = 6
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Solving for y2:
- y2 = 2 * ym - y1
- y2 = 2 * 3 - 5
- y2 = 1
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The Unknown Endpoint: Therefore, the coordinates of the other endpoint (x2, y2) are (6, 1).
Conclusion
Finding the unknown endpoint of a line segment, given one endpoint and its midpoint, is a straightforward process that relies on the fundamental midpoint formula. By substituting the known values into the formula and solving for the unknowns, we can accurately determine the coordinates of the missing endpoint. This method has applications in various fields, including geometry, coordinate geometry, and computer graphics, where understanding the relationships between endpoints and midpoints is crucial for analyzing and manipulating geometric shapes.
