Proof 5.12
Using coordinates, write a detailed step-by-step proof that the set of points equidistant from the two fixed points A and B is the perpendicular bisector of the segment AB.
Let be a line segment with end points and . Denote as and as
1. Using the midpoint formula, we can determine that the midpoint is located at .
2. Consider a fourth point equidistant from and . Denote as .
3. We can determine the distance from to using the distance formula.
4. Similarly, we can determine the distance from to .
5. Since we know that is equidistant from and , we can set the formulas from 3 and 4 equal to one another and solve for .
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6. Since the x-coordinate of is zero, we know that it lies on y-axis. Recall, also lies on the y-axis which is perpendicular by definition to the x-axis where and are located.
7. Since is the midpoint of and lies on a line perpendicular to , the line is the perpendicular bisector of .
Therefore, the set of points equidistant from two fixed points is the perpendicular bisector of the segment formed by the fixed points.