How to find an endpoint with midpoint and one endpoint?
Understand the Problem
The question is asking for the method to find an unknown endpoint of a line segment when given its midpoint and one of its endpoints. To solve this, we can use the midpoint formula, which states that the midpoint M of a line segment with endpoints A(x1, y1) and B(x2, y2) is given by M = ((x1 + x2)/2, (y1 + y2)/2). We can rearrange this to solve for the unknown endpoint.
Answer
The unknown endpoint \(B\) can be found using \(x_2 = 2x_m - x_1\) and \(y_2 = 2y_m - y_1\).
Answer for screen readers
The coordinates of the unknown endpoint (B) are given by (B(x_2, y_2)), where: $$ x_2 = 2x_m - x_1 $$ $$ y_2 = 2y_m - y_1 $$
Steps to Solve
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Identify given values Identify the coordinates of the midpoint (M(x_m, y_m)) and the known endpoint (A(x_1, y_1)).
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Write the midpoint formula Recall the midpoint formula: $$ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) $$ where (M) is the midpoint, (A) is one endpoint, and (B(x_2, y_2)) is the unknown endpoint.
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Set up equations Equate the coordinates from the midpoint formula to those provided: $$ x_m = \frac{x_1 + x_2}{2} $$ $$ y_m = \frac{y_1 + y_2}{2} $$
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Solve for unknown endpoint coordinates Rearrange these equations to solve for (x_2) and (y_2): $$ x_2 = 2x_m - x_1 $$ $$ y_2 = 2y_m - y_1 $$
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Substitute known values Now, substitute the values of (x_m), (y_m), (x_1), and (y_1) into the equations to find (x_2) and (y_2).
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Final coordinates of unknown endpoint The coordinates (B(x_2, y_2)) will be your unknown endpoint.
The coordinates of the unknown endpoint (B) are given by (B(x_2, y_2)), where: $$ x_2 = 2x_m - x_1 $$ $$ y_2 = 2y_m - y_1 $$
More Information
This is a common method in geometry for determining an unknown point based on its relation to other points on a line segment. Understanding the midpoint formula is fundamental in coordinate geometry and helps to visualize relationships between points.
Tips
- Incorrectly using the midpoint formula: Ensure you understand which point is the midpoint and which is the known endpoint.
- Neglecting to isolate the variable correctly: When rearranging the equations for (x_2) and (y_2), pay close attention to the algebra to avoid sign mistakes.
